1 / 22

Function Composition

Function Composition. Function Composition. Fancy way of denoting and performing SUBSTITUTION But first ….let’s review. Function notation: f(x) This is read “f of x”. This DOES NOT MEAN MULITPLY f and x. Finding the Value of a Function. If f(x) = 3x - 1, find f(2).

glenda
Download Presentation

Function Composition

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Function Composition

  2. Function Composition • 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) = x2 - 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 - 4x2 + 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: [fog](x) or f(g(x)) means “f of g of x”. [g0f](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. • f(g(3)) • g(f(3)) • f(g(0)) • g(f(0)) • 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. • f(g(3)) • First find g(3). Remember this means find y when x = 3. When x = 3 y = -1 Now find f(-1). What is y when x = -1 for the f-function? 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) = x2 + 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 = x2 – 8x + 16 +x - 4 = x2 – 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) = x2 + 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(x2 + x ) = x2 + 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

More Related