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Section 12.1 Composite and Inverse Functions

Section 12.1 Composite and Inverse Functions. Phong Chau. 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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Section 12.1 Composite and Inverse Functions

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  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? • 10% off and $20 off

  3. Domain of f || Domain of g Range of g Range of f g f x g(x) f(g(x)) 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 “fcompose 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) (x 2– 5x) x = 2 – 3 = 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? • If F = 50 , 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. A picture • Let F(C) = 9/5 C +32. Then Domain of f = range of f inverseDomain of f inverse = range of f f-1 = C(F) 0 41 5 10 32 20 59 68 50 15 f =F(C)

  12. Formal Definition • If f and g be two functions such that (f◦g)(x) = x and (g◦f)(x) = xthen 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

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