in pre-calculus 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 uh basic parent functions let's start with the graph yal 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 X could 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 now this function will continue to go in this direction forever it will keep going to the right and it will go up so that's why uh X 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 all the possible y values viewing the graph from the bottom to the top the lowest y value is negative 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're writing the domain and range in inval notation now let's move on to our next parent function y is equal to x^2 you've seen this in quadratic equations and the shape is that of a parabola it's an upward you 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 yv value of this function is zero it will never be below zero it can never be negative anytime you square a number it's always going to be positive unless you square zero which is zero the highest y value value is infinity therefore the range is from0 to infinity and since it includes zero you need to use a bracket the next parent function that we have coming up is a cubic function y is equal to X Cub what do you think the general shape of this graph looks like this function is an increas in function 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 polom function next up we have the square root of x so this function increases at a decrease in 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 zero and the highest is infinity so the domain is from 0 to Infinity now it includes zero because if you plug in zero the square root of 0 is zero looking at the range the lowest y value is zero the highest is infinity so the range is also from zero 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 of4 is not a real number the cube root of8 is equal to -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 vshape 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 z numbers now what about the range notice that the range is restricted the lowest y value for this particular function is zero the highest is infinity so the range is from zero 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/x so what's the parent shape for this function for this particular function you need to realize that there is a horizontal ASM toote known as Y is equal to Z and is also a vertical ASM toote which is X is equal to Z 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 ASM toote from the domain notice that X could be anything except zero there's no yellow line on this vertical ASM toote so you have to remove the vertical asmil from a domain so the domain is going to be negative Infinity to Z Union Z to Infinity so this means that X could be anything except zero now whenever you have a problem with a horizontal ASM toote like this one where there's only one vertical ASM toote notice that the function can never have a yvalue of zero it doesn't touch the horizontal line so you need to remove the horizontal ASM toote from the range so this is also going to be negative Infinity to 0 Union 0 to Infinity here's another rational function 1 /x^2 so this graph is similar to the last graph but there are some differences there's still going to be a vertical ASM toote at xal 0 and we're still going to have a horizontal ASM toote which is y is equal to zero now because the function is squared it can only have positive values which means the function has to be above zero it can 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 xaxis going to be here so this particular function is symmetric about the Y AIS as you can see the right side looks like a mirror image of the left side so it's a reflection across the y AIS 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 ASM toote so once again we need to to remove the vertical ASM toote from the uh domain so X cannot be zero 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 zero but the highest is positive infinity and because the horizontal ASM toote is y equal 0 the function never touches the horizontal ASM toote so it's greater than zero but not equal to zero 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 zero but you'll never reach a y-v value of zero if you plug in one into X Y is going to be one if you increase it to let's say 100 Y is going to be 0.1 if you increase it to a th000 y is going to be 0.0000 01 it's going to get closer and closer to zero but it never actually touches zero so the range is going to be from zero to Infinity but it does not include zero since that's a horizontal ASM tone so we have to use the parenthesis by the way if you want to find more of my pre-calculus videos check out the website that I have below video- tr.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 ASM toote at y equal 0 so the graph is going to start from the x-axis now this function increas inrees at an increase in rate so it looks like this the domain of this function is all real numbers X can be anything now if we look at the y- values the lowest y value is zero the highest is infinity but it does not include zero so the range is z 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 ASM toote at xal 0 the natural log function has a vertical ASM toote at xal 0 I meant to say that the exponential function had a horizontal ASM at y equal 0 and for this one it's xal 0 because they're inverses of each other everything is switched the horizontal as ASM toote of the exponential function is now the vertical ASM toote of the natural log function now this function is going to increase but at a decrease in 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 uh original function is the range of the inverse you can switch them so as you can see the lowest x value is zero 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 X and here's the natural log function lnx notice that it is symmetrical across the line yal 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 yals 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 sign graph sign is a sooso function that repeats over time it's a periodic function it starts from the origin and if you have the positive sign 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 sign which is a one so it's going to vary from one to negative 1 negative sign 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 s function is negative Infinity to Infinity however the range has limitations as you can see the lowest y value is Nega 1 and the highest is one if the amplitude is one sometimes that number can change but for this particular function the range is1 to one 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 cosine starts below the X AIS and then it goes up the amplitude varies from the amplitude is one in this case which means the Y values vary from netive 1 to 1 so the range is the same as a sign 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 trick function I'm going to go over in this video and that is the tangent function tangent has the vertical ASM toote attive pi/ 2 and at pi/ 2 and it's an increase in function now this shape repeats across each vertical ASM the next vertical ASM is at 3 Pi 2 and then it's going to have the the same shape so X could be anything except the vertical ASM tootes so X cannot be n pi/ 2 for this exact function where N is a number such as plus or minus one plus or min-2 Plus or -3 and it goes on forever actually not two it's only odd numbers so plus to- one plus to- 3 plus to- 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 ofx now if we put a two in front of it what effect will it have on a graph putting the two on the outside will stretch the graph vertically so the Y values will double so putting the two in front you could say the function was stretched vertically by a factor of two likewise if you put a half in front of f ofx it's going to be a vertical shrink by a factor of two so all of the Y values will be half of what they were the X values will remain the same you just got to reduce all the y- values by half now what about putting a two on the inside what effect will that have on the graph this is going to be a horizontal shrink by a factor of two 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 1/2 on the inside if we replace x with 12 x what effect will that have on the graph so this is going to stretch horizontally by a factor of two so all of the X values will double but the Y values remain the same so it stretches horizontally now what about this one f ofx - 4 if you set the inside term x - 4 equal to Z and if you solve for x you'll see that X is equal to four so what this tells us is that the graph is going to shift four 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 four units to the right that's it now what about f of x + 3 what effect will that have on a graph so in this case it's going to shift left three units so it's going to be somewhere over here okay that looks a little bit bigger so let me uh try that again there we go so it's a horizontal shift left three 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 AIS 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 would 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 AIS 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 it's 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 Gonna open upward so with that in mind it's going to be going facing in the upward Direction and the long side is going to be towards the left side so reflected 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 at -2 -3 and this point is at1 2 and the next point is at 32 let's say this is 5g5 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 -23 will now be -3 -2 which let's put marks in a graph -3 -2 is over here now this one is going to be instead of being netive 1 comma 2 it's now 21 so 21 is over here let's connect those two points with a line now the next one is going to be 2 comma 3 which is uh over here and then the last point will be -5 comma 5 which is uh somewhere up here so this graph is the inverse function which it reflects over the line yal X now if you recall we said that the graph y = x^2 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^2 minus 3 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 = x^2 the domain will always be all real numbers now what about the range what is the lowest yvalue that you see the lowest is3 and the highest is infinity so therefore the range is from -3 to Infinity let's try another example try this one x^2 + 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 X2 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 R numbers there's no restrictions on the value that you can plug in for X so what is the um 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 two so the range is from negative Infinity to positive2 let's try this one y = x - 2 rais the 3 power the parent function is X cub 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 zero and solve for x so the new origin is located at positive2 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 /x looks like this there was a vertical ASM toote at xal 0 but now it's xus three on the bottom so it's going to shift three units to the right if you want to find a new vertical ASM toote set the denominator equal to zero so the New Vertical ASM toote has been shifted three units to the right so it's over here the horizontal ASM toote is still y equal 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 asmo so the range is going to be from negative Infinity to infin I mean to zero Union 0 to Infinity now the domain to be anything except the vertical ASM toote so the domain for this graph is infinity to 3 Union 3 to Infinity what 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 ASM toote will no longer be yal 0 it's going to be yal 2 so let's go ahead and graph the function so the vertical ASM toote is zero that is x equal Z but the new horizont Al Asm toote is now Y = 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 ASM toote and to write the range we need to take out the horizontal asmt so it's going to be from netive 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 + 2 equal to Z this will give us the vertical ASM toote which is uh -2 this number tells us the horizontal ASM toote 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 ASM toote which is at -2 and the horizontal ASM toote which is at three now the graph won't be here because we have a negative sign it's going to reflect over the horizontal asmt 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 ASM it's going to be there so to write the domain of the function we just need to remove the vertical Asm toote so it's going to be from negative Infinity to -2 Union -2 to infinity and to write the range we see that the lowest y value is negative Infinity the highest is infinity however it's not going to have a yvalue of three so we got to remove the horizontal asot 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 - 2^ 2 + 3 now if you recall because it's squared it's going to Wi to be uh symmetrical about the y- AIS but in this case it's really going to be symmetrical about the the vertical ASM toote because notice that the vertical ASM toote is at xal 2 now it's been shifted two units to the right so it's over here and we can see that the horizontal ASM toote is y equal 3 it's been shifted up three 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 ASM toote so therefore to write the domain it's going to be everything except the vertical ASM toote of two and the range we can see that the lowest y value is three but it doesn't include the horizontal asm2 at y equal 3 and the highest y value is infinity so the range is from three to Infinity let's try another example like this one so let's say if it's A1 overx + 3^ 2 minus 2 so go ahead and take a minute to graph this particular function so the vertical ASM toote is x equal to3 so it's been shifted three units to the left the horizontal ASM toote is -2 specifically Y is equal to -2 so it has been shifted two units down now the negative sign tells us is that it's going to reflect over the horizontal ASM toote 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 the second quadrant it's going to be in the third quadrant as well so it still reflects it's still like um a reflection over the uh vertical asmo but this negative caused it to reflect over the Blue Line the horizontal asmt but it's still symmetrical over the vertical asmo 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 ASM from it and for the range we can see that the lowest y value is negative infinity and the highest is the horizontal ASM toote of -2 so the range is going to be from negative Infinity to -2 but it does not include -2 now what about this function so we know that the absolute value of x is a vshape that opens upward now for this one it's going to shift three units to the right and it's going to shift one unit up so the new origin is at 3 comma 1 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 R numbers however the range can vary the lowest yvalue is one and the highest is infinity so the range is going to be 1 to infinity and it includes one so we need to use a bracket so go ahead and try this example what if we have two - x - 2 so the graph shifts two units to the right if you set x - 2 equal to Z x is 2 it also shifts up two units so the new origin is at 2 comma 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 two 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 x + 2 if you recall e to the X was a function that goes up like this and it had a horizontal ASM toote of y equal 0 now the graph has been shifted two units up so the new horizontal ASM toote is y is equal to 2 so we should plot that first so the graph is going to start from the horizontal ASM toote 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 yvalue is two the highest is infinity so the range is going to be from two to infinity and since it starts from an ASM toote it does not include two now let's try a natural log function if you recall Ln X has a vertical ASM toote at x equal Z but now it's been shifted three units to the right so the New Vertical ASM toote is that x equals 3 and then the shape is going to be the same it's going to increase at a decrease in rate now let's write the domain the lowest x value is three 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 uh trig functions how can we graph 2 sinx + 1 now let's review the sign function sign starts at the center it goes up then it goes down and then it goes back to the center the amplitude is one 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 one the second thing is the amplitude is two so from this Center Line it's going to go up two and also down two so if you add two to one the highest point will be three and if you subtract two from one the lowest point is going to be negative 1 so it's going to vary between negative 1 and three 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 s 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 is1 the highest is three so that's the range for this particular function let's try another example let's try -3 cosine x + 4 so the first thing we should realize is that the center of the graph shifts four units up -3 + 4 will give us the lowest value of 1 3 + 4 will give us the highest value of seven 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 could extend the graph if you want so the domain is going to be the same our R numbers negative Infinity to Infinity but the range is 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 root function it's going to be all real 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 could 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 uh quadrant one one way I like to think about it is that X is positive and Y is positive X and Y are positive in quadrant one here's quadrant 2 3 and four now for this function it reflects over the xaxis 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 two so it reflects over the Y AIS 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 -2 3 now we need to know where the graph is going to go is it going to go towards quadrant 1 towards quadrant 2 towards quadrant 3 or towards Quadrant 4 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 uh 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 -2 and the highest is infinity because it can keep going to the right so the domain is -2 to infinity and it includes -2 now the range the lowest yalue is negative Infinity and the highest is three so the range is from netive Infinity to three and it includes three if you plug in -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 zero if you add X to both sides X is five so it shifts five units to the right and it's going to shift up three units so the new origin is located at 5 comma 3 now will it shift towards quadrant 1 or towards quadrant 2 3 or four 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 POS so it's going to be from Infinity to five now what about the range as we travel towards the left this will continually decrease it's going to decrease slower and slower but it's still it can go down forever so the lowest y value is negative infinity and the highest is what we see here three so the range is going to be from negative Infinity to three so that covers uh 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 ofx is x² + 3 and G of X is 2x - 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 ofx is x^2 + 3 but what we're going to do is we're going to replace x with G so let's replace x with 2x - 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 - 4^ 2 is the same as 2x - 4 * 2x - 4 so we need to foil 2x * 2x that's uh 4x^2 and 2x * -4 is -8x and this is also uh 8X and finally -4 * -4 is pos6 so now let's combine like terms so this is going to be 4x^2 we can combine these two8 + 8 is -6 and 16 + 3 is 19 so this is f of G of X now what about G of f ofx so this time we want to take F and insert it into G so let's begin by writing the function uh for G but don't plug in anything into X so we're going to replace x with x^2 + 3 so all we need to do right now is distribute the two 2 * x^2 is 2x^2 2 * 3 is 6 and 6 - 4 is 2 so this is the composite function let's try another example let's say f ofx is 3x - 5 and G of X is XB - 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 of2 so we can use this function let's replace x with two 2 to the 3 power which is 2 * 2 * 2 three times that's 8 and 8 - 9 is1 so G of 2 is equal to1 so therefore we can replace G of 2 with1 so now let's take that value and plug it into this equation so f of1 is going to be 3 * -1 - 5 which is -3 - 5 and that's equal to8 so this is the answer now let's try this one g of f of three so this time we're going to find the value of f of three first using this equation so plugging three into the equation we have 3 * 3 which is 9 and 9 - 5 is 4 so F of 3 is four so let's replace it with four so now we're looking for G of four and so let's plug it into that equation so G of 4 is equal to 4 3r - 9 4 Cub is 64 and 64 - 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 ofx is equal to 7 x - 3 what is the inverse function of F ofx how can we find it whenever you wish to find an inverse function first replace f ofx with uh y y and F ofx are the same next switch X and Y and then finally solve for y so let's add three to both sides so x + 3 is equal to 7 Y and next let's divide both sides by 7 so x + 3 over 7 is equal to Y and that's the inverse function so the inverse function is x + 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 and g of X will equal x if they're inverses of each other and also G of f ofx 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 into F that is let's replace this x with x + 3 7 so it's going to be 7 * x + 3 7 - 3 so we can see that the sevens will cancel and so it's going to be x + 3 - 3 3 - 3 is 0o 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 got to show that g of f ofx 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 - 3 so -3 + 3 is z so that leaves us with 7 x / x 7 / 7 is 1 1 * X is just X so now you know how to prove two functions um if they're inverses of each other let's try one more example let's say f ofx is equal to X2 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 ofx with Y next let's switch X and Y and then we'll have to solve for y so we need to take this 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 a value of f of G of X so let's call this G ofx so let's take this and plug in here so it's going to be X2 but we're going to replace x with the square root of x which the squ otk of x^ 2 is basically rootx * rootx which isun x^2 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 X2 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 equal 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 yal X so that is it for this video If you like this video feel free to uh subscribe you can click the little red button that you see there also if you want to find more videos on pre-calculus just check out my website video- t.net and you could find videos on pre-cal chemistry physics calculus which you might be taking next year so if you want to get a head start you can look at that as well so thanks for watching