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**Section 12.1 Composite and Inverse Functions**

Phong Chau

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**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? 10% off and $20 off

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**|| 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)

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**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)

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**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

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

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(a)

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(b)

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

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**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”

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**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)

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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)

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

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**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.

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Examples Find the inverse of the following functions

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