Understanding Function Notation and Examples

Hi everybody, today's lesson is about function notation. The first few slides are important, they don't require my intervention at all here in terms of speaking, but they are fundamental to your understanding of function notation and what it is and what it means. So please do give it the time, even though I'm not speaking over the slides, please do give them your attention. and make sure you understand the definitions of what it is and what the notation means. Thank you. So here on this slide, we're just pointing out how these sorts of questions would have been structured in the past. The old way we would have asked, well, here's an equation, y equals x squared plus 1. Find the value of y when x is equal to 2. And we would have subbed in 2 for x. 2 squared is 4 plus 1 is 5. So y would equal 5. Using function notation, we would ask the same question in this following way. f of x is equal to x squared plus 1. Find f at 2. And then we would state the solution as f at 2 is equal to 5. So both of those are the exact same question, but we're going to use this new way of function notation going forward in mathematics in grade 11 and 12. Okay, so in this example, which I believe is number one in your notes, we're given a function g of x, and it's a quadratic, it's 2x squared minus 3x plus 1. So we know it is a function, and we are asked in part a to find g at negative 2. So what does that mean? It means we're being asked to find the value of our function, the y value. when x is equal to negative 2. So g at negative 2 is equal to 2 times negative 2 squared minus 3 times negative 2 plus 1. and so we have 2 times 4 plus 6 plus 1 uh that equals 15 right a plus 6 plus 1 so we have g at negative 2 is equal to 15 in this case and we can also conduct operations so over here in b We're asked to find g at 4 and subtract g at 2, right? So the value of our function when g is 4, that particular value, we're asked to take that and subtract from it the value of our function when x is 2. So what we're being asked to do here is take g at 4. So 2 times 4 squared minus 3 times 4 plus 1 and subtract the value of g at 2. So we've got 2 times 2 squared minus 3 times 2 plus 1. And then we can just simplify the algebra here. So 32 minus 12 plus 1. minus 8 minus 6 plus 1. So we've got 21 minus, I do the brackets first, 8 minus 6 is 2 plus 1 is 3. So we're subtracting 3. And so the value of g at 4 minus g at 2. is 21 minus 3, which is 18. In example C, what we're being asked to do here is take or find g at negative 5, and once we're finished doing that, we're going to multiply it by 3. So we're looking for 3 times the value of my function when x is equal to negative 5. So 3 times g at, actually we don't need that bracket, 3 times g at negative 5 is equal to 3 times 2 times negative 5 squared minus 3 times negative 5 plus 1. So that's 3 times 50. plus 15 plus 1 and so we've got 3 times 66 which gives us 198. In example D we're asked to find G at T. Well, so what is the value of our function when x is actually equal to t? Well, it's just 2 times t squared minus 3t plus 1. And there's nothing else we can do because we don't know the value of t. All right, so in this way, we can use our function notation to evaluate our function at certain values. of x and to find and therefore find our y values right continuing on with the same question the same function we're asked to find g at t plus 5 so this is equal to 2 times t plus 5 squared minus 3 times t plus 5 plus 1. So we have two times, we just FOIL that out, t squared plus 10 t plus 25 minus 3 t minus 15 plus 1. So we have 2 t squared plus 20 t plus 25 minus 3 t minus 15 plus 1. and gather our like terms so we have 2 t squared 20 t so that's plus 17 t and we've got 25 minus 15 is 10 plus 1 is 11. in part f we're asked to find what our what the value of the function is when x is 1 find that value and then subtract 3 from it So we're asked to do the following. 2 times 1 squared minus 3 times 1 plus 1 minus 3. So this is 2 minus 3 plus 1 minus 3. 2 minus 3 is negative 1 plus 1 is 0. So the answer is negative 3. In part g, we are asked to find x if g of x happens to be 21. So if g of x is 21, then the left side of our equation is 21, is equal to 2x squared minus 3x plus 1. So we're just being asked to solve a quadratic. So 0 is equal to 2x squared minus 3x plus 1. minus 21. And so this is the equation that we need to solve. Right, so in order to solve these types of equations, we know that we need to factor, if we can, or use our quadratic formula. Somehow we need to get at what x equals. So does this factor? Yes, it does. It factors to 2x plus 5, x minus 4, and therefore x is equal to negative 5 over 2, or x is equal to 4. All right, so you may, in some of the questions, be asked to, you know, reach back to grade 10 and sort out some of your quadratics. We can also be given a graph and be asked to work things out using function notation. This just goes back to your fundamental idea of what function notation is from those first few slides and that'll let you know how to deal with this in terms of a graph right. So in part a some of these are not included on your student notes that's okay they're fairly quick we'll go through them all. F at one we just go to our graph for an x value of one. We're up here at approximately 5.5, right? So the y value, that's my y value. When x is 1, in this case, is 5 and a half. When x is 0, right, then f at 0, the value of my function, when x is 0, is up here at 7. For C, this is asking what value of X gives me a Y value of zero, right? And where is my Y value zero on my graph? There's only one spot and it's over here at 10. So X must equal 10 in order for my Y value to be zero. Similarly, in part D. what value of x will make my y value 3 my y value of 3 occurs right about here so at a value of oh we can say 2.75 maybe it's going to be a bit of a guess right approximately 2.75 in part e What is the value of my function at an x value of 12? Well the graph only goes up to 10, an x value of 10, so we have no idea. So we will say that that does not exist, right? The graph could exist at 12, but we're not shown anything about any values of x, or the values of our function for x values greater than 10, so we can state that it does not exist. determine when f of x is equal to x squared. So when does the value of my function equal the value of x squared? Well, we can take a look at zero. What's zero squared? Zero squared is zero, right? So Do we have a point down here at 0, 0? No, we don't. So the value of my function when x is 0 does not equal 0 squared. So it's not at 0. What about 1? At 1, 1 squared is 1. So my function does not exist at 1, 1. There's nothing here. So 1 is not a solution. What about 2? At an x value of 2, my function is at 4. That is x squared, right? That is 2 squared. So we know that one possibility is that x is equal to 2. And we can go along and check all of the others. So at an x value of 3, does my function equal 9? No. 4, does it equal 16? No. 5, 25, and so on? No. So x equals 2 is going to be the only. Option that satisfies the condition here where the value of my function is equal to the value of x squared. All right, in example three from your student notes, given f of x is a linear function, find the equation if f at three is equal to four and f at zero is equal to negative two. Again, if this and this. are confusing please go back and take a look at the first number of slides in the explanation of what function notation is we're really going back and focusing on the definitions of what things are and how they work in order to work our way through this type of question so we know it's a linear function so we know it's going to have an arrangement or an equation that looks like y equals n x plus b And then what is this information here on the right hand side that I've underlined? Well, we know that f at 3 is equal to 4. Therefore, we know that our line goes through the point 3, 4. And we also know that our line goes through the point 0, negative 2. And so since we have two points, we can create the equation of a line based on how we've done it for the last two years in grade 9 and 10. So we'll get slope is equal to negative 2 minus 4 over 0 minus 3. So we've got y is equal to 2x plus b. And then we'll sub in one of the points to find our b value. So negative 2 is equal to... 2 times 0 plus b. Negative 2 is equal to 0 plus b and b is equal to negative 2. Therefore, the equation is y is equal to 2x minus 2. All right, so there we put our ideas of linear functions, knowing what the properties of linear functions are, and our skills with straight lines, together with our ideas of function notation. All right, in the next few examples we're going to work with some interesting equations and some interesting, putting some interesting twists on some of the questions. So we'll find g at x plus 2. So g of x is root x plus 4. So we're going to need this. x is not x. It's x plus 2 plus 4. So this is root x plus 6. So in that example, the x, right, we subbed in the x plus 2 for the x in our equation. In part b, f at h of x. What does that mean? Well, it means that the x value in equation f. right so this x and this x here aren't going to just be replaced by a number they're actually going to be replaced by this whole algebraic expression h of x so we have x squared but x isn't x anymore x is 4x minus 12 and that's squared right from here We have to subtract x, which is now h of x, right? So we're subtracting 4x minus 12, and then we can simplify. So we have 16x squared minus 96x plus 144. Minus 4x plus 12. We end up with 16x squared minus 96x minus 100x plus 44 plus 12 is plus 156. Okay, so this is an example where we replaced x in the original equation with an algebraic expression instead of just a number. But we did the same things. We replaced the x in the equation here, which was f, and then we simplified. The next example, 2 times f of x minus h of x. So we have 2 times x squared minus x, that's f of x, minus h of x, which is 4x minus 12. And so we have 2x squared minus 2x minus 4x plus 12 is equal to 2x squared minus 6x plus 12. All right. This one looks pretty interesting. Let's take a look and work our way through it. So we have h at x plus 1 minus h at x divided by 2. So one step at a time, we have h of x is 4x minus 12. We're going to replace the x with x plus 1. So we get 4 times x plus 1 minus 12. That's the first part. Now we're going to subtract h of x, 4x minus 12. Let's keep the brackets the same type, and we're going to divide everything by 2. So we have 4x plus 4 minus 12 minus 4x plus 12. all over two all right we got uh 4x minus 4x and then we have 4 minus 12 is negative 8. okay i guess the 12s right 12s are going to divide going to cancel as well but uh so we have negative 8 here plus 12 which is 4 over 2 right so that expression at the top simplifies down to just being two all right in this example there is a minor typo here in your in your notes i believe i believe in your notes this says just x but it should be negative x Please make that change in your notes, otherwise the resulting quadratic is not going to factor. So here we have f of x is equal to g at negative x squared. So what this is saying is that x squared minus x is equal to the square root of negative x plus 4 squared. So we have x squared minus x. The squaring undoes the square root, and we are left with negative x plus 4. And we're asked to determine x. So we have to solve this for x. If we move all the terms over to the left-hand side, we get x squared minus 4 is equal to 0. And you can see that's a difference of squares. So we have x plus not 4, but 2. x plus 2 x minus 2 is equal to 0 and therefore x is equal to plus or minus 2. Okay so I think that finishes off the examples. I really really really cannot stress enough how important those introductory slides are. You must know and understand what function notation is, what its purpose is. and what each of the pieces means. For example, there was a slide that went over what does this piece mean? What does that X in the bracket mean? And then we also went over what does this whole thing mean? What does it represent? Those are the important things to know before you start trying to answer some of the homework questions. Okay and as usual the homework is listed on the notes and it's on your outline as well.

and make sure you understand the definitions of what it is and what the notation means. Thank you. So here on this slide, we're just pointing out how these sorts of questions would have been structured in the past.

The old way we would have asked, well, here's an equation, y equals x squared plus 1. Find the value of y when x is equal to 2. And we would have subbed in 2 for x. 2 squared is 4 plus 1 is 5. So y would equal 5. Using function notation, we would ask the same question in this following way. f of x is equal to x squared plus 1. Find f at 2. And then we would state the solution as f at 2 is equal to 5. So both of those are the exact same question, but we're going to use this new way of function notation going forward in mathematics in grade 11 and 12. Okay, so in this example, which I believe is number one in your notes, we're given a function g of x, and it's a quadratic, it's 2x squared minus 3x plus 1. So we know it is a function, and we are asked in part a to find g at negative 2. So what does that mean? It means we're being asked to find the value of our function, the y value.

when x is equal to negative 2. So g at negative 2 is equal to 2 times negative 2 squared minus 3 times negative 2 plus 1. and so we have 2 times 4 plus 6 plus 1 uh that equals 15 right a plus 6 plus 1 so we have g at negative 2 is equal to 15 in this case and we can also conduct operations so over here in b We're asked to find g at 4 and subtract g at 2, right? So the value of our function when g is 4, that particular value, we're asked to take that and subtract from it the value of our function when x is 2. So what we're being asked to do here is take g at 4. So 2 times 4 squared minus 3 times 4 plus 1 and subtract the value of g at 2. So we've got 2 times 2 squared minus 3 times 2 plus 1. And then we can just simplify the algebra here. So 32 minus 12 plus 1. minus 8 minus 6 plus 1. So we've got 21 minus, I do the brackets first, 8 minus 6 is 2 plus 1 is 3. So we're subtracting 3. And so the value of g at 4 minus g at 2. is 21 minus 3, which is 18. In example C, what we're being asked to do here is take or find g at negative 5, and once we're finished doing that, we're going to multiply it by 3. So we're looking for 3 times the value of my function when x is equal to negative 5. So 3 times g at, actually we don't need that bracket, 3 times g at negative 5 is equal to 3 times 2 times negative 5 squared minus 3 times negative 5 plus 1. So that's 3 times 50. plus 15 plus 1 and so we've got 3 times 66 which gives us 198. In example D we're asked to find G at T. Well, so what is the value of our function when x is actually equal to t? Well, it's just 2 times t squared minus 3t plus 1. And there's nothing else we can do because we don't know the value of t.

All right, so in this way, we can use our function notation to evaluate our function at certain values. of x and to find and therefore find our y values right continuing on with the same question the same function we're asked to find g at t plus 5 so this is equal to 2 times t plus 5 squared minus 3 times t plus 5 plus 1. So we have two times, we just FOIL that out, t squared plus 10 t plus 25 minus 3 t minus 15 plus 1. So we have 2 t squared plus 20 t plus 25 minus 3 t minus 15 plus 1. and gather our like terms so we have 2 t squared 20 t so that's plus 17 t and we've got 25 minus 15 is 10 plus 1 is 11. in part f we're asked to find what our what the value of the function is when x is 1 find that value and then subtract 3 from it So we're asked to do the following. 2 times 1 squared minus 3 times 1 plus 1 minus 3. So this is 2 minus 3 plus 1 minus 3. 2 minus 3 is negative 1 plus 1 is 0. So the answer is negative 3. In part g, we are asked to find x if g of x happens to be 21. So if g of x is 21, then the left side of our equation is 21, is equal to 2x squared minus 3x plus 1. So we're just being asked to solve a quadratic.

So 0 is equal to 2x squared minus 3x plus 1. minus 21. And so this is the equation that we need to solve. Right, so in order to solve these types of equations, we know that we need to factor, if we can, or use our quadratic formula. Somehow we need to get at what x equals.

So does this factor? Yes, it does. It factors to 2x plus 5, x minus 4, and therefore x is equal to negative 5 over 2, or x is equal to 4. All right, so you may, in some of the questions, be asked to, you know, reach back to grade 10 and sort out some of your quadratics. We can also be given a graph and be asked to work things out using function notation.

This just goes back to your fundamental idea of what function notation is from those first few slides and that'll let you know how to deal with this in terms of a graph right. So in part a some of these are not included on your student notes that's okay they're fairly quick we'll go through them all. F at one we just go to our graph for an x value of one. We're up here at approximately 5.5, right? So the y value, that's my y value.

When x is 1, in this case, is 5 and a half. When x is 0, right, then f at 0, the value of my function, when x is 0, is up here at 7. For C, this is asking what value of X gives me a Y value of zero, right? And where is my Y value zero on my graph?

There's only one spot and it's over here at 10. So X must equal 10 in order for my Y value to be zero. Similarly, in part D. what value of x will make my y value 3 my y value of 3 occurs right about here so at a value of oh we can say 2.75 maybe it's going to be a bit of a guess right approximately 2.75 in part e What is the value of my function at an x value of 12? Well the graph only goes up to 10, an x value of 10, so we have no idea.

So we will say that that does not exist, right? The graph could exist at 12, but we're not shown anything about any values of x, or the values of our function for x values greater than 10, so we can state that it does not exist. determine when f of x is equal to x squared. So when does the value of my function equal the value of x squared?

Well, we can take a look at zero. What's zero squared? Zero squared is zero, right? So Do we have a point down here at 0, 0?

No, we don't. So the value of my function when x is 0 does not equal 0 squared. So it's not at 0. What about 1?

At 1, 1 squared is 1. So my function does not exist at 1, 1. There's nothing here. So 1 is not a solution. What about 2?

At an x value of 2, my function is at 4. That is x squared, right? That is 2 squared. So we know that one possibility is that x is equal to 2. And we can go along and check all of the others.

So at an x value of 3, does my function equal 9? No. 4, does it equal 16? No. 5, 25, and so on?

No. So x equals 2 is going to be the only. Option that satisfies the condition here where the value of my function is equal to the value of x squared. All right, in example three from your student notes, given f of x is a linear function, find the equation if f at three is equal to four and f at zero is equal to negative two. Again, if this and this.

are confusing please go back and take a look at the first number of slides in the explanation of what function notation is we're really going back and focusing on the definitions of what things are and how they work in order to work our way through this type of question so we know it's a linear function so we know it's going to have an arrangement or an equation that looks like y equals n x plus b And then what is this information here on the right hand side that I've underlined? Well, we know that f at 3 is equal to 4. Therefore, we know that our line goes through the point 3, 4. And we also know that our line goes through the point 0, negative 2. And so since we have two points, we can create the equation of a line based on how we've done it for the last two years in grade 9 and 10. So we'll get slope is equal to negative 2 minus 4 over 0 minus 3. So we've got y is equal to 2x plus b. And then we'll sub in one of the points to find our b value.

So negative 2 is equal to... 2 times 0 plus b. Negative 2 is equal to 0 plus b and b is equal to negative 2. Therefore, the equation is y is equal to 2x minus 2. All right, so there we put our ideas of linear functions, knowing what the properties of linear functions are, and our skills with straight lines, together with our ideas of function notation.

All right, in the next few examples we're going to work with some interesting equations and some interesting, putting some interesting twists on some of the questions. So we'll find g at x plus 2. So g of x is root x plus 4. So we're going to need this. x is not x. It's x plus 2 plus 4. So this is root x plus 6. So in that example, the x, right, we subbed in the x plus 2 for the x in our equation. In part b, f at h of x.

What does that mean? Well, it means that the x value in equation f. right so this x and this x here aren't going to just be replaced by a number they're actually going to be replaced by this whole algebraic expression h of x so we have x squared but x isn't x anymore x is 4x minus 12 and that's squared right from here We have to subtract x, which is now h of x, right?

So we're subtracting 4x minus 12, and then we can simplify. So we have 16x squared minus 96x plus 144. Minus 4x plus 12. We end up with 16x squared minus 96x minus 100x plus 44 plus 12 is plus 156. Okay, so this is an example where we replaced x in the original equation with an algebraic expression instead of just a number. But we did the same things. We replaced the x in the equation here, which was f, and then we simplified.

The next example, 2 times f of x minus h of x. So we have 2 times x squared minus x, that's f of x, minus h of x, which is 4x minus 12. And so we have 2x squared minus 2x minus 4x plus 12 is equal to 2x squared minus 6x plus 12. All right. This one looks pretty interesting. Let's take a look and work our way through it.

So we have h at x plus 1 minus h at x divided by 2. So one step at a time, we have h of x is 4x minus 12. We're going to replace the x with x plus 1. So we get 4 times x plus 1 minus 12. That's the first part. Now we're going to subtract h of x, 4x minus 12. Let's keep the brackets the same type, and we're going to divide everything by 2. So we have 4x plus 4 minus 12 minus 4x plus 12. all over two all right we got uh 4x minus 4x and then we have 4 minus 12 is negative 8. okay i guess the 12s right 12s are going to divide going to cancel as well but uh so we have negative 8 here plus 12 which is 4 over 2 right so that expression at the top simplifies down to just being two all right in this example there is a minor typo here in your in your notes i believe i believe in your notes this says just x but it should be negative x Please make that change in your notes, otherwise the resulting quadratic is not going to factor. So here we have f of x is equal to g at negative x squared.

So what this is saying is that x squared minus x is equal to the square root of negative x plus 4 squared. So we have x squared minus x. The squaring undoes the square root, and we are left with negative x plus 4. And we're asked to determine x. So we have to solve this for x. If we move all the terms over to the left-hand side, we get x squared minus 4 is equal to 0. And you can see that's a difference of squares.

So we have x plus not 4, but 2. x plus 2 x minus 2 is equal to 0 and therefore x is equal to plus or minus 2. Okay so I think that finishes off the examples. I really really really cannot stress enough how important those introductory slides are. You must know and understand what function notation is, what its purpose is. and what each of the pieces means.

For example, there was a slide that went over what does this piece mean? What does that X in the bracket mean? And then we also went over what does this whole thing mean?

What does it represent? Those are the important things to know before you start trying to answer some of the homework questions. Okay and as usual the homework is listed on the notes and it's on your outline as well.

Transcript for:Understanding Function Notation and Examples

Transcript for:
Understanding Function Notation and Examples