In PreCalculus, one of the most important topics that you need to know is functions. How to graph functions, how to find the domain and range. So in this video I'm going to give you a basic understanding of the different types of functions you need to know, how the graphs look like, and also how to find the domain and range of these functions.
But let's go over some basic parent functions. Let's start with the graph y equals x. Now this graph is basically a linear function.
The line should pass through the origin. And for any linear function, the domain is negative infinity to infinity. The domain represents the x values that the function can have.
As you can see, f is the value of f. X can be anything. Whenever you're analyzing the domain of the function, look at the function from left to right. The lowest X value that it can have is negative infinity, and the highest is infinity. will continue to go in this direction forever.
It will keep going to the right and it will go up. So that's why x can be anything. You can plug in any value into x.
So the domain represents all real numbers. Now the range is associated with the y values. To find the range of the function, look at all the possible y values. View the graph from the bottom to the top. The lowest y value is negative infinity.
infinity because this portion of the graph keeps going down. The highest y value is positive infinity, so the range is negative infinity to infinity. Anytime you have an infinity symbol you should always use a parentheses never brackets with it whenever you write in the domain and range and interval notation now let's move on to our next parent function y is equal to x squared You've seen this in quadratic equations, and the shape is that of a parabola. It's an upward u-shape. So that's the parent function.
What do you think the domain and range of this function is? So what is the lowest x value? x can be anything.
It can range from negative infinity to infinity. So for any quadratic function, The domain is always going to be all real numbers. Now what about the range? What is the lowest y value that you see? The lowest y value of this function is 0. It will never be below 0, it can never be negative.
Anytime you square a number, it's always going to be positive, unless you square 0, which is 0. The highest y value is infinity. Therefore, the range is from 0 to infinity. And since it includes 0, you need to use a bracket. The next parent function that we have coming up is a cubic function, y is equal to x cubed.
What do you think the general shape of this graph looks like? This function is an increasing function. It looks like this.
Now what is the domain and range of the function? So the lowest x value is negative infinity, it can keep moving towards the left. And the highest x value is infinity. So the domain includes all real numbers, x can be anything. Now, analyzing the range, we can see that the lowest y-value is negative infinity and the highest is infinity.
So the range also varies from negative infinity to infinity, which is true of any cubic polynomial function. Next up, we have the square root of x. So this function increases at a decreasing rate.
It looks like this. So, based on this shape, what do you think the domain of this function is going to be? What is the lowest x value that you see?
The lowest x value is 0, and the highest is infinity. So, the domain is from 0 to infinity. Now, it includes 0, because if you plug in 0, the square root of 0 is 0. Looking at the range, the lowest y-value is 0, the highest is infinity.
So the range is also from 0 to infinity. Now, it can vary as well, based on... the transformations of this graph, but for this particular parent function, that's the domain and range. Now what about the cube root of x?
What is the shape of this function? The cube root of x looks like the square root of x, but it's on both sides. It's symmetric about the origin, so it looks like this. Let me see if I can draw that better. The domain is from negative infinity to infinity.
You can plug in any x value into the function. For the square root of x, you can't plug in a negative value. The square root of negative 4 is not a real number.
The cube root of negative 8 is equal to negative 2. That's a real number. Now the range, the lowest y value is negative infinity, and the highest is infinity. So y can be anything as well. Now what is the shape of the absolute value of x function? What's the parent function?
This function is basically a graph that forms a v-shape and it opens upward. x can vary from negative infinity to infinity. You can plug in anything and replace it with x, or replace x with that number.
So therefore, the domain of an absolute value function is all real numbers. Now what about the range? notice that the range is restricted. The lowest y-value for this particular function is 0, the highest is infinity.
So the range is from 0 to infinity. We're going to go over some more examples with transformations later, but for now let's just review the basic parent functions. So next we're going to have a rational function 1 over x. So, what's the parent shape for this function? For this particular function, you need to realize that there is a horizontal asymptote, known as y is equal to 0, and there's also a vertical asymptote, which is x is equal to 0. I'll talk about how to find that later, but for now, just know that it exists.
This graph looks like this. It's symmetric about the origin. Here's the origin.
As you can see, this side is a reflection of that side across this point. Now whenever you have a rational function, you need to remove the vertical asymptote from the domain. Notice that x could be anything except 0. There's no yellow line on this vertical asymptote. So you have to remove the vertical asymptote from the domain.
So the domain is going to be negative infinity to 0 union 0 to infinity. So this means that x could be anything except 0. Now whenever you have a problem with a horizontal asymptote like this one, where there's only one vertical asymptote, notice that the function can never have a y value of 0. It doesn't touch the horizontal line. So you need to remove the horizontal asymptote from the range.
So this is also going to be negative infinity to 0 union 0 to infinity. Here's another rational function, 1 over x squared. So this graph is similar to the last graph, but there are some differences. There's still going to be a vertical asymptote at x equals 0. And we're still going to have a horizontal asymptote, which is y is equal to 0. Now, because the function is squared, it can only have positive values, which means the function has to be above 0. It can't be negative. So the right side is going to look the same.
The left side, it won't be here. Instead, it's going to flip over the x-axis. It's going to be here. So this particular function is symmetric about the y-axis.
As you can see, the right side looks like a mirror image of the left side. So it's a reflection across the y-axis. Now what is the domain of this function?
As we can see, the lowest x value is negative infinity, and the highest is infinity. But, there's no point on the vertical asymptote. So, once again, we need to remove. the vertical asymptote from the domain.
So x cannot be 0. So it's going to be negative infinity to 0, union 0 to infinity. Now what about the range? Notice that the lowest y value is 0, but the highest is positive infinity.
And because the horizontal asymptote is y equals 0, the function never touches the horizontal asymptote. So it's greater than 0 but not equal to 0. There's no value of x that you can plug in that will give you a value of y. You can get very close to 0, but you'll never reach a y value of 0. If you plug in 1 into x, y is going to be 1. If you increase it to, let's say, 100, y is going to be.0001.
If you increase it to 1,000, y is going to be.00001. It's going to get closer and closer to 0, but it never actually touches 0. So the range is going to be from 0 to infinity, but it does not include 0 since that's a horizontal asymptote. So we have to use a parenthesis.
By the way, if you want to find more of my pre-calculus videos, check out the website that I have below, video-trudeau.net. You can find more videos on pre-calculus, even a playlist on calculus, algebra, trigonometry, if you need help on those. General chemistry organic chemistry and even physics so feel free to take a look at that when you get a chance But now let's continue now.
What is the parent function of an exponential function e to the x? What is the general shape of this function the first thing you need to realize is that it has a horizontal asymptote at y equals 0 So the graph is going to start from the x-axis now this function increases at an increasing rate. So it looks like this. The domain of this function is all row numbers. X can be anything.
Now if we look at the y values, the lowest y value is 0, the highest is infinity, but it does not include 0. So the range is 0 to infinity. Now the next one we're going to go over is the natural log function. It turns out that the natural log function is the inverse function of an exponential function.
The exponential function had a horizontal asymptote at x equals 0. The natural log function has a vertical asymptote at x equals 0. I meant to say that the exponential function had a horizontal asymptote at y equals 0. And for this one, it's x equals 0. Because they're inverses of each other, everything is switched. The horizontal asymptote of the exponential function is now the vertical asymptote of the natural log function. Now this function is going to increase, but at a decreasing rate.
Also, the domain of the exponential function is the range of the natural log function. And the range of the exponential function is the domain of the natural log function. Whenever you have two inverse functions, the domain of the... original function is the range of the inverse.
You can switch them. So as you can see the lowest x value is 0, the highest is infinity. So therefore the domain is 0 to infinity.
Now looking at the range, the lowest y value is negative infinity and the highest is infinity. This graph will keep going up forever. It just goes up slowly.
So it's negative infinity to infinity. Notice what happens when we put the two functions together. So here's the exponential function, e to the x, and here's the natural log function, ln x. Notice that it is symmetrical across the line y equals x. Each graph is a reflection across this line.
And that is typical of inverse functions. An inverse function is, or will reflect across the line y equals x compared to its original function. So that's something you want to keep in mind.
Now let's move on into trig functions. What is the general shape of the sine graph? Sine is a sinusoidal function that repeats over time. It's a periodic function.
It starts from the origin. And if you have the positive sine function, it's going to go up, and then it's going to repeat like a wave. So I'm just going to draw one cycle. The amplitude is the number in front of sine, which is a 1. So it's going to vary from 1 to negative 1. Negative sine is very similar.
However... It's going to start from the origin, but instead of going up, it's going to go down. So as you can see, it reflects over the x-axis.
Now this function continues forever in both directions. Therefore, the domain for any sine function is negative infinity to infinity. However, the range has limitations.
As you can see, the lowest y value is negative 1, and the highest is 1, if the amplitude is 1. Sometimes that number can change. But for this particular function, the range is negative 1 to 1. Now what about the graph y is equal to positive cosine x? Cosine starts at the top, and then it forms a sine wave. It repeats, but if you just want to draw only one cycle, it looks like this. Negative cosine starts below the x-axis, and then it goes up.
The amplitude varies from... The amplitude is 1 in this case, which means the y values vary from negative 1 to 1. So the range is the same as the sine function. And because it's a periodic function that goes forever, the domain is all real numbers. Negative infinity to infinity. And there's one more trig function I'm going to go over in this video, and that is the tangent function.
Tangent has the vertical asymptote at negative pi over 2 and at pi over 2. And it's an increase in function. Now this shape repeats across each vertical asymptote. The next vertical asymptote is at 3 pi over 2. and then it's going to have the same shape.
So x could be anything except the vertical asymptotes. So x cannot be n pi over 2 for this exact function, where n is a number such as plus or minus 1, plus or minus 2, plus or minus 3, and it goes on forever. Actually, not two.
It's only odd numbers. So plus or minus 1, plus or minus 3, plus or minus 5. And then the pattern repeats. The range of a tangent function is negative infinity to infinity. To write the domain is quite difficult because this is going to go on forever.
So x can be anything except these values. Now let's review transformations. So let's say if this is the parent function, and it's going to have a shape that looks like this. Let's call it f of x.
Now if we put a 2 in front of it, what effect will it have on the graph? Putting a 2 on the outside will stretch the graph vertically, so the y values will double. So, putting the 2 in front, you could say the function was stretched vertically by a factor of 2. Likewise, if you put a half in front of f of x, it's going to be a vertical shrink by a factor of 2. So, all of the y values will be half of what they were.
The x values remain the same. You just got to reduce all the y values by half. Now what about putting a 2 on the inside? What effect will that have on the graph?
This is going to be a horizontal shrink by a factor of 2. So the y values will be the same, but the x values will be half of what they were. So it shrinks horizontally. Let's erase just that.
Now what if we put a 1 half on the inside? If we replace x with 1 half x? What effect will that have on the graph?
So this is going to stretch horizontally by a factor of 2. So all of the x values will double, but the y values will remain the same. So it stretches horizontally. What about this one?
f of x minus 4. If you set the inside term x minus 4 equal to 0, and if you solve for x, you'll see that x is equal to 4. So what this tells us is that the graph is going to shift 4 units to the right. So it's going to look something like this. It's going to have the same shape, it's just it shifted 4 units to the right. That's it. Now what about f of x plus 3?
What effect will that have on a graph? So in this case, it's going to shift left 3 units. So it's going to be somewhere over here.
Okay, that looks a little bit bigger, so let me try that again. There we go. So it's a horizontal shift, left 3 units.
Now what if we put a negative on the outside of the function? What effect will it have? In this case, it's going to reflect over the x-axis.
So it's going to flip. So it's going to look something like this. Now what if we put a negative on the inside of the function? How will the graph transform? In this case, it's going to reflect over the y-axis.
So it's going to look like that. Now here's a challenge one. What if we have a negative on the outside and on the inside?
How will that affect a graph? To answer this question... you need to realize that this is a combined effect of putting the negative on the outside and putting it on the inside.
So let's review what we did. When the negative was on the outside, it reflected over the x-axis. So it looked something like that. And when a negative was on the inside, it reflected over the y-axis.
And so it looked like this. This means it's going to reflect over the origin, which is a combination of reflecting over the x and the y axis. So somehow we have to combine these two graphs. So the first thing we need to realize is that this long side... is going to be on the left side, as we see here.
The second thing is this graph is not going to face in a downward direction. It's going to open upward. So with that in mind, it's going to be facing in the upward direction, and the long side is going to be towards the left side.
So reflect it over the origin. And so that's how you can do it. And finally, if you see this, this simply means it's the inverse function. So let's say if... I'm going to use points in this case.
So let's draw our original shape. let's say this point is that negative 2 negative 3 and this point is that negative 12 The next point is at 3 2 let's say this is 5 Negative 5 and that's f of x to graph the inverse function All you need to do is switch the x and the y values So this point negative 2 negative 3 will now be negative 3 negative 2 which Let's put marks on the graph Negative 3, negative 2 is over here. Now, this one is going to be, instead of being negative 1, 2, it's now 2, negative 1. So, 2, negative 1 is over here.
Let's connect those two points of the line. now the next one is going to be 2 comma 3 which is over here and then the last point will be negative 5 comma 5 which is somewhere up here so this graph is the inverse function which reflects over the line y equals x Now, if you recall, we said that the graph y equals x squared is a parabola that opens in the upward direction that looks like this. So now, what if we have a different graph that is associated with that parent function? Let's say x squared minus string.
How can we graph this function? This minus 3 is a vertical shift. It's going to bring the function down 3 units. But the graph is still going to open in the upward direction. So it's going to look like this.
So for this particular function, what is the domain and the range? We can see that the lowest x value is negative infinity and the highest is positive infinity. For any type of quadratic function or parabolic function that looks like this, y equals x squared, the domain will always be all real numbers. Now what about the range?
What is the lowest y-value that you see? The lowest is negative 3, and the highest is infinity. So therefore the range is from negative 3 to infinity.
Let's try another example. Try this one. Negative x squared plus 2. So go ahead and draw a rough sketch of the graph and also determine the domain and range of the function.
So this graph is going to shift up two units, and it's not going to open upward because there's a negative in front of the x squared. So it's going to reflect over the x-axis, meaning that it's going to open in a downward direction. The domain of this function is going to be the same, all real numbers.
There's no restrictions on the value that you can plug in for x. So what is the range for this function? What is the lowest y value, and what is the highest y value? The lowest is negative infinity. The highest is 2. So the range is from negative infinity to positive 2. Let's try this one.
y is equal to x minus 2 raised to the third power. The parent function is x cubed. And we know the general shape for this function.
It looks something like this. So therefore, how can we graph this function? What type of transformation do we have?
If you recall from the rules that we just went over, this means that the function is going to move two units to the right. If you ever forget, set the inside equal to 0 and solve for x. So the new origin is located at positive 2. So this graph is going to look like this.
It simply shifted two units to the right. For any cubic function, the domain will remain negative infinity to infinity, and the range is going to stay the same, negative infinity to infinity. X and Y can be anything.
Now what if we have a rational function that looks like this? How can we graph it? So if you recall, the graph 1 over x It looks like this. There was a vertical asymptote at x equals 0, but now it's x minus 3 on the bottom, so it's going to shift 3 units to the right. If you want to find the new vertical asymptote, set the denominator equal to 0. So the new vertical asymptote has been shifted 3 units to the right, so it's over here.
The horizontal asymptote is still y equals 0. So the graph looks like this now. It's simply been shifted three units to the right. So what is the range and the domain of this function?
Let's start with the range. The range could be anything except the horizontal asymptote. So the range is going to be from negative infinity to zero, union zero to infinity.
Now, the domain can be anything except the vertical asymptote. So, the domain for this graph is negative infinity to 3, union 3 to infinity. What if the transformation is on the outside? Let's say if it's not part of the fraction. This number will cause the graph to shift two units up, so the horizontal asymptote will no longer be y equals 0, it's going to be y equals 2. So let's go ahead and graph the function.
So the vertical asymptote is 0, that is x equals 0, but the new horizontal asymptote... Is now y equals 2 the shape of the graph will still be the same It's simply been shifted two units up so to write the domain of the function once again, we just need to remove the vertical asymptote and to write the range we need to take out the horizontal asymptote so it's going to be from negative infinity to 2 union 2 to infinity so now what if we have a combination of transformations and reflections try this one so if we set x plus 2 equal to 0 This will give us the vertical asymptote, which is negative 2. This number tells us the horizontal asymptote, which is y is equal to 3. And the negative sign tells us that it reflects over the x-axis, which for rational functions, it's equivalent to reflecting over the y-axis. So let's begin by plotting the vertical asymptote, which is at negative 2. and the horizontal asymptote, which is that 3. Now the graph won't be here, because we have a negative sign. It's going to reflect over the horizontal asymptote.
So it's going to look like this. And in the last example, we had a graph, the curve was here, but now it's going to be on the other side of the horizontal asymptote. It's going to be there. So, to write the domain of the function, we just need to remove the vertical asymptote. So, it's going to be from negative infinity to negative 2, union negative 2 to infinity.
And to write the range, we can see that the lowest y value is negative infinity, the highest is infinity. However, it's not going to have a y value of 3, so we've got to remove the horizontal asymptote. So it's from negative infinity to 3, union 3 to infinity.
Now let's work on some more examples. Go ahead and try this one. 1 over x minus 2 squared plus 3. Now if you recall, because it's squared, it's going to want to be...
symmetrical about the y-axis but in this case is really going to be symmetrical about the the vertical asymptote because notice that the vertical asymptote is at x equals 2 now it's been shifted two units to the right so it's over here And we can see that the horizontal asymptote is y equals 3. It's been shifted up 3 units. So the graph is going to be on this side, but it won't be here. Because it's squared, the left side and the right side will be the same across the vertical asymptote. So therefore, to write the domain, it's going to be everything except the vertical asymptote of 2. And the range, we can see that the lowest y value is 3, but it doesn't include the horizontal asymptote at y equals 3. And the highest y value is infinity. So the range is from 3 to infinity.
Let's try another example like this one. So let's say if it's negative 1. over x plus 3 squared minus 2. So go ahead and take a minute to graph this particular function. So the vertical asymptote is x is equal to negative 3. So it's been shifted three units to the left. The horizontal asymptote is negative 2. Specifically, y is equal to negative 2. So, it's been shifted 2 units down.
Now, the negative sign tells us that it's going to reflect over the horizontal asymptote. So, it won't be in the first quadrant. It's going to be in the fourth quadrant.
And it's not going to be in the second quadrant. It's going to be in the third quadrant as well. So, it still reflects.
It's still like a... a reflection of the vertical asymptote, but this negative caused it to reflect over the blue line, the horizontal asymptote, but it's still symmetrical over the vertical asymptote, that's what I meant to say. So that's how the graph looks like.
To write the domain, it's going to be the same, all we need to do is take away the vertical asymptote from it. And for the range, we can see that the lowest y value is negative infinity, and the highest is the horizontal asymptote of negative 2. So the range is going to be from negative infinity to negative 2, but it does not include negative 2. Now what about this function? So we know that the absolute value of x...
is a V shape that opens upward. Now for this one, it's going to shift 3 units to the right, and it's going to shift 1 unit up. So the new origin is at, and it's going to open in the same direction.
For any absolute value function, the domain is going to be the same. It's all real numbers. However, the range can vary. The lowest y value is 1, and the highest is infinity.
So the range is going to be 1 to infinity, and it includes 1. So we need to use a bracket. So go ahead and try this example. What if we have 2 minus x minus 2? So the graph shifts two units to the right.
If you set x-2 equal to 0, x is 2. It also shifts up two units. So the new origin is at 2,2. But notice that we have a negative sign in front of the absolute value function.
So instead of opening in the upward direction, it's going to open downward. So it's going to look like this. The domain will be the same.
It's going to be from negative infinity to infinity. But for the range, we can see that the lowest y value is negative infinity, and the highest is 2. So it's going to be from negative infinity to 2. So that's it for absolute value functions. Now let's move on to exponential functions.
Let's say if we have the function e to the x plus 2. If we recall... Etx was a function that goes up like this and it had a horizontal asymptote of y equals 0 now The graph has been shifted two units up, so the new horizontal asymptote is y is equal to 2. So we should plot that first. So the graph is going to start from the horizontal asymptote, and it's going to increase. The domain for any exponential function is all real numbers.
X can be anything. However, the range has restrictions. The lowest y value is 2, the highest is infinity.
So the range is going to be from 2 to infinity, and since it starts from an asymptote, it does not include 2. Now let's try a natural log function. If you recall, ln x has a vertical asymptote at x equals 0, but now it's been shifted 3 units to the right. So the new vertical asymptote is that x equals 3. And then the shape is going to be the same.
It's going to increase at a decreasing rate. Now, let's write the domain. The lowest x value is 3. The highest is infinity.
So it's going to be from 3 to infinity. Now, if we analyze the range, the lowest y value is negative infinity. And the highest is positive infinity.
So the range is all row numbers. y can be anything. Now let's try some examples with trig functions.
How can we graph 2 sine x plus 1? Now let's review the sine function. Sine starts at the center, it goes up, then it goes down, and then it goes back to the center.
The amplitude is 1. But in this case, it's going to be different. First, the graph is going to shift one unit up. So the center point is no longer the x-axis. The center point is at 1. The second thing is the amplitude is 2. So from this center line, it's going to go up 2 and also down 2. So if you add 2 to 1, the highest point will be 3. And if you subtract 2...
From 1, the lowest point is going to be negative 1. So it's going to vary between negative 1 and 3. So it's going to start at the top, I mean at the center. It's going to go up, then back to the middle, then to the bottom, and then back to the center. So this is one cycle of the sine wave. The domain for any sine or cosine function is always going to be negative infinity to infinity.
Because it can keep going on forever. However, the range... is based on these values. We can see that the lowest y value is negative 1, the highest is 3. So that's the range for this particular function. Let's try another example.
Let's try negative 3 cosine x plus 4. So the first thing we should realize is that the center of the graph shifts 4 units up. Negative 3 plus 4 will give us the lowest value of 1. 3 plus 4 will give us the highest value of 7. Negative cosine starts at the bottom. It's going to go to the middle, then to the top, back to the middle, and then back to the bottom. So it's going to look something like this.
And you can extend the graph if you want. So the domain is going to be the same. Our row number is negative infinity to infinity. But the range... It's based on these values.
It's going to be from 1 to 7. What about this one? The general shape of the cube root of x looks like this. So the origin, which was 0, 0, is now going to move up one unit.
So the graph now looks like this. The domain for any cube root function is going to be R root numbers, negative infinity to infinity. And the range is the same. So you really don't have to worry about this function very much. X and Y can be anything.
But now let's focus on the square root of X. I want you to graph these four functions. The square root of positive x is going to shift towards quadrant 1. One way I like to think about it is that x is positive and y is positive.
x and y are positive in quadrant 1. Here's quadrant 2, 3, and 4. Now for this function, it reflects over the x-axis. If you look at the signs, x is positive, y is negative. x is positive towards the right, y is negative. below, so that takes us towards quadrant 4. And since it reflects over the x-axis, it's going to look like this.
Now here, x is negative and y is positive. x is negative towards the left, y is positive as you go up, so this is going to go towards quadrant 2. So it reflects over the y-axis, so it looks like this. And the last one, we have a negative on the inside and on the outside.
So when x is negative, it goes to the left, and when y is negative, it goes down. So this is going to reflect over the origin, and so it's going to look like this. Make sure you're aware of these four shapes.
So now let's work on some problems. Go ahead and graph this particular function, and write the domain and range in interval notation. So first, we need to find out where the new origin is located. It's been shifted two units left, and up three. So it's located at negative two, three.
Now we need to know where the graph is going to go. Is it going to go towards quadrant one, towards quadrant two, towards quadrant three, or towards quadrant four? So the first thing that we can see is that x is positive.
So it's going to go towards the right, either towards quadrant 1 or towards quadrant 4. But notice that y is negative, so it can't go up, it has to go down. So it's going to go down towards quadrant 4, where x is positive and y is negative. So it's going to go to the right and down. So that's the parent function, or just a rough sketch of... this function.
So now let's find the domain and range. Let's focus on the x values. The lowest x value is negative 2 and the highest is infinity because it can keep going to the right.
So the domain is negative 2 to infinity and it includes negative 2. Now the range, the lowest y value is negative infinity and the highest is 3. So the range is from negative infinity to 3, and it includes 3. If you plug in negative 2 into x, y will equal 3. So that's the domain and range for this function. Let's try this one. So the first thing that we can do is set the inside equal to 0. If you add X to both sides, X is 5. So it shifts 5 units to the right, and it's going to shift up 3 units.
So the new origin is located at 5, 3. Now, will it shift towards quadrant 1, or towards quadrant 2, 3, or 4? So first, there's a negative sign in front of X, so it's going to go to the left. That means it's going to be one of these two.
So it's not going to go towards the right. And there's a negative sign in front of the square root, so you can view it as y being negative. So it's not going to be this one, it's going to go down.
So our rough sketch looks like this. So now we can write the domain. The lowest x value is negative infinity, and the highest is what we see inside, positive 5. So it's going to be from negative infinity to 5. Now what about the range? As we travel towards the left, this will continually decrease.
It's going to decrease lower and slower, but still, it can go down forever. So the lowest y-value is negative infinity, and the highest is what we see here, 3. So the range is going to be from negative infinity to 3. So that covers radical and square root functions. Now let's move on to a new topic, that is the composition of functions. You'll see this in pre-calculus quite often. Let's say that f of x is x squared plus 3, and g of x is 2x minus 4. If you see something that looks like this, what would you do?
Where g is inside of f. So this is known as a composite function, where a function is inside of another function. To find this value, insert g into f. So f of x is x squared plus 3. But what we're going to do is we're going to replace x with g.
So let's replace x with 2x minus 4. So that's how you can find a composite function f of g of x. But typically, you may have to simplify this result. 2x minus 4 squared is the same as 2x minus 4 times 2x minus 4. So we need to FOIL. 2x times 2x, that's 4x squared.
And 2x times negative 4 is negative 8x. And this is also negative 8x. And finally, negative 4 times negative 4 is positive 16. So now let's combine like terms. So this is going to be 4x squared. We can combine these two.
Negative 8 plus negative 8 is negative 16. And 16 plus 3 is 19. So this is f of g of x. Now what about g of f of x? So this time, we want to take f and insert it into g. So let's begin by writing the function for g, but don't plug in anything into x.
So we're going to replace x with x squared plus 3. So all we need to do right now is distribute the two. 2 times x squared is 2x squared, 2 times 3 is 6, and 6 minus 4 is 2. So this is the composite function. Let's try another example.
Let's say f of x is 3x minus 5, and g of x is x cubed minus 9. Go ahead and evaluate the function f of g of 2. So what would you do? In the last example, there was an x here, so we got a function in terms of x. If there's a number, your final answer should equal a number. In this case, the first thing we can do is find g of 2, so we can use this function.
Let's replace x with 2. 2 to the third power, which is 2 times 2 times 2, 3 times, that's 8. And 8 minus 9 is negative 1, so g of 2 is equal to negative 1. So therefore, we can replace g of 2 with negative 1. So now let's take that value and plug it into this equation. So f of negative 1 is going to be 3 times negative 1 minus 5, which is negative 3 minus 5, and that's equal to negative 8. So this is the answer. Now let's try this one, g of f of 3. So this time, we're going to find the value of f of 3 first, using this equation.
So plug in 3 into the equation, we have 3 times 3, which is 9, and 9 minus 5 is 4. So f of 3 is 4. So let's replace it with 4. So now we're looking for g of 4, and so let's plug it into that equation. So g of 4 is equal to 4 to the 3rd minus 9. 4 cubed is 64. And 64 minus 9 is 55. So the final answer, the whole thing, is equal to 55. The next topic on our list is finding the inverse function. So let's say that f of x is equal to 7x minus 3. What is the inverse function of f of x?
How can we find it? Whenever you wish to find an inverse function, first replace f of x with y. y and f of x are the same. Next, switch x and y. And then finally, solve for y.
So let's add 3 to both sides. So x plus 3 is equal to 7y. And next, let's divide both sides by 7. So, x plus 3 over 7 is equal to y, and that's the inverse function.
So, the inverse function is x plus 3 over 7. Now, if you're given two functions, and if you wish to see if they're inverses of each other, here's what you can do. Let's call the inverse function g of x. It turns out that f of g of x will equal x if they're inverses of each other.
And also, g of f of x should equal x if they're inverses. Let's prove that these two functions are indeed inverses of each other. So let's start with f of g of x. So let's take g and insert it into f.
That is, let's replace this x with x plus 3 over 7. So it's going to be 7 times x plus 3 over 7 minus 3. So we can see that the 7s will cancel. And so it's going to be x plus 3 minus 3. 3 minus 3 is 0. So we do get x, which tells us that they're inverses of each other. But you need to prove the other equation as well.
You've got to show that g of f of x is also equal to x. So this time we're going to take f and insert into g. So let's replace this x with 7x minus 3. So negative 3 plus 3 is 0. So that leaves us with 7x divided by x. 7 divided by 7 is 1. 1 times x is just x. So now you know how to prove two functions if they're inverses of each other.
Let's try one more example. Let's say f of x is equal to x squared with a restricted domain. So we only want the right side of the function. Go ahead and find the inverse function.
So let's replace f of x with y. Next, let's switch x and y. And then we'll have to solve for y. So we need to take the square root of both sides.
So the square root of x is equal to y. So therefore, we can say the inverse function is the square root of x. Now, to prove that they're inverses of each other, we can find the value of f of g of x. So let's call this g of x.
So let's take this and plug in here. So it's going to be x squared, but we're going to replace x with the square root of x, which the square root of x squared is basically root x times root x, which is root x squared, and that's simply equal to x. And if you do it the other way, it's going to be the same. So you can see that they're inverses of each other.
But now let's graph these two functions. The right side of x squared looks like this. Let me use a different color. And the square root of x looks like that. As you can see, these two functions are symmetrical.
about the line y equals x. We can extend this further. So as you can see, two functions that are inverses of each other will always reflect across the line y equals x. So that is it for this video. If you like this video, feel free to subscribe.
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