1. Function Composition

2. Fancy way of denoting and performing SUBSTITUTION But first ….let’s review.

3. Function notation: f(x) This is read “f of x”. This DOES NOT MEAN MULITPLY f and x.

4. Finding the Value of a Function If f(x) = 3x - 1, find f(2). Substitute 2 for x and simplify. f(2) = 3(2) – 1 = 6 - 1 = 5

5. Finding the Value of a Function Given g(x) = x 2 - x, find g(-3) g(-3) = (-3) 2 - (-3) = 9 - -3 = 9 + 3 = 12 g(-3) = 12 Substitute -3 in for each x then simplify.

6. Finding the Value of a Function Given g(x) = 3x - 4x 2 + 2, find g(5) g(5) = 3(5) - 4(5) 2 + 2 = 15 - 4(25) + 2 = 15 - 100 + 2 = -83 g(5) = -83 Put 5 in place of every x. Simplify.

7. Find the Value of a Function Given f(x) = x - 5, find f(a+1) f(a + 1) = (a + 1) - 5 = a+1 - 5 f(a + 1) = a - 4

8. Function Composition Notation: [f o g](x) or f(g(x)) means “f of g of x”. [g 0 f](x) or g(f(x)) means “g of f of x” The notation in orange is typically used.

9. Find the Value of a Composition of Functions Given f(x) = 2x + 5 and g(x) = 8 + x, find f(g(-5)). First find g(-5)…… g(-5) = 8 + -5 = 3. Now put 3 in place of g(-5) (Plug 3 into the f function) f(g(-5)) = f(3) = 2(3) + 5 = 6 + 5 = 11

10. Find the Value of a Composition of Functions Given f(x) = 2x + 5 and g(x) = 8 + x, find g(f(-5)). (Note how this problem is different from the problem on the previous slide.) First find f(-5) by plugging -5 in for x in the f function. f(-5) = 2(-5) + 5 = -10 + 5 = 5 Now find g(f(-5)) by plugging 5 in for x in the g function. g(5) = 8 + 5 = 13. g(f(-5)) = 13

11. Use the graphs of y = f(x) and y = g(x) to find each of the following compositions. a.f(g(3)) b.g(f(3)) c.f(g(0)) d.g(f(0)) e.g(g(3))

12. Use the graphs of y = f(x) and y = g(x) to find each of the following compositions. Find f(g(3)) First find g(3). Remember this means find y when x = 3 for the g function. When x = 3 y = -1

13. Use the graphs of y = f(x) and y = g(x) to find each of the following compositions. a.f(g(3)) First find g(3). Remember this means find y when x = 3. When x = 3 Now find f(-1). What is y when x = -1 for the f- function? y = -1 When x = -1 y = -5 So, f(g(3)) = -1

14. Use the graphs of y = f(x) and y = g(x) to find each of the following compositions. Find: g(f(3)) First find f(3). What is y when x = 3 on the f function? f(3) = 3

15. Use the graphs of y = f(x) and y = g(x) to find each of the following compositions. Find: g(f(3)) First find f(3). What is y when x = 3 on the f function? f(3) = 3 Now find g(3). What is y when x = 3 on the g- function? g(3) = -1 So g(f(3)) = -1

16. Use the graphs of y = f(x) and y = g(x) to find each of the following compositions. Find: f(g(0)) f(g(0)) = f(2) = 4 Find: g(f(0)) g(f(0)) = g(0) = 2 Find: g(g(3)) g(g(3)) = g(-1) = 3

17. Composition of Functions Function Composition is just fancy substitution, very similar to what we have been doing with finding the value of a function. The difference is that instead of plugging in a number we will be plugging in another function.

18. Composition of Functions You still replace x with whatever is in the parentheses. f(g(x)) basically means plug the g function into the f function and simplify. g(f(x)) basically means plug the f function into the g function and simplify.

19. Composition of Functions If f(x) = x 2 + x and g(x) = x - 4, find f(g(x)) Plug the g function (x – 4) into the f function in place of every x and simplify. f(g(x)) = f(x - 4) = (x - 4) 2 + x - 4 = x 2 – 8x + 16 +x - 4 = x 2 – 7x + 12

20. Given f(x) = 2x + 2 and g(x) = x + 4, find f(g(x)). f(g(x + 4)) = 2(x +4) +2 = 2x + 8 + 2 = 2x + 10 final answer

21. If f(x) = x 2 + x and g(x) = x – 4 find g(f(x)). This time put the f-function into the g-function in place of x. g(f(x)) =g(x 2 + x ) = x 2 + x – 4 Final answer.

22. If f(x) = 2x + 2 and g(x) = x + 4, find g(f(x)). g(f(x)) = 2x +2 +4 = 2x + 6