1. Section 12.1 Composite and Inverse Functions
Phong Chau

2. Examples Convert temperature from K to F
From K to C, we have C = K – 273
From C to F, we have F = 1.8 C + 32
 What is the formula converting from K to F directly? That is, what is the function F in terms of K?
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3. || g f x g(x) f(g(x)) Domain of f Domain of g Range of g Range of f
Domain of f(g)
Range of f(g)
(f ◦ g)(x) = f(g(x))
Or f(g)

4. Composition of Functions
Given two functions f and g, the composite function is defined by
(f ◦ g)(x) = f(g(x))
Read “f compose g of x”
Note: In general (f ◦ g)(x) ≠ (g ◦ f)(x)

5. Composition of Functions
Let f(x) = 2x – 3 and g(x) = x 2 – 5x. Determine (f ◦ g)(x).
(f ◦ g)(x) = f(g(x)) = f(x 2 – 5x)
= – 3
(x 2 – 5x) x
= 2x 2 – 10x – 3

6. Example
Let and g(x) = x2 – 2.
Determine
(a) (f ◦ g)(x)
(b) (g ◦ f)(x)

7. (a)

8. (b)

9. Example
Let and
Determine
(f ◦ g)(x)
(g ◦ f)(x)

10. Example Temperature Function If F = 68 , what is C?
How can you find a formula that convert temperature from F back to C?
Do you see how these 2 formulas “undo” each other?
 They are “inverse functions”

11. Let F(C) = 9/5 C +32. Then A picture C 5 10 15 20 F(C) 32 41 50 59 68
Domain of f = range of f inverse Domain of f inverse = range of f C 5 10 15 20
F(C) 32 41 50 59 68
f-1 = C(F) 41 5 10 32 20 68 59 15 50
f =F(C)

12. Formal Definition
If f and g be two functions such that (f◦g)(x) = x and (g◦f)(x) = x then we say the function g is the inverse of the function f and the function g is denoted by f -1.
NOTE: f -1 ≠ 1 / f(x)

13. Examples
Verify that the two functions are inverses of each other.

14. Finding the Inverse of a Function
Recall: The relationship of two functions that are inverses is that for any coordinate (x, y) on one, the other has the coordinate (y, x).
To find an inverse algebraically:
1. Replace f (x) with y.
2. Interchange x and y in the equation.
3. Solve the new equation for y.
4. Rename it f -1.

15. Examples
Find the inverse of the following functions