Presentation on theme: "Section 12.1 Composite and Inverse Functions"— Presentation transcript:
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 – 273From 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 || 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)
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 fC5101520F(C)3241505968f-1 = C(F)41510322068591550f =F(C)
12 Formal DefinitionIf 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 ExamplesVerify 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 ExamplesFind the inverse of the following functions