Sec. 2.7 Inverse Functions

Inverse of a function Found by interchanging the coordinates in each ordered pair Denoted by f -1

Inverse functions f(x) : {(1,5), (2,6), (3,7), (4,8)} Note: the domain of a f is equal to the range of f -1 and vice versa

Inverse functions undo each other Inverse functions undo each other. When you form the composite of inverse functions you get the identity function

Write the inverse function of f(x) : {(3,5), (-2,1), (-5,3), (-4,-1)} f -1(x): {(5,3), (1,-2), (3,-5), (-1,-4)}

Graph the f(x) and f -1(x) we just found Note: the graph f -1is a reflection of the graph of f over the line y=x

To determine if 2 functions are inverses Graph – if they are a reflection over line y=x then they are inverses Algebraically – if (f ◦ f -1)(x)=(f -1 ◦f)(x)=x f(f -1(x))= f -1(f(x))=x then they are inverses

Example f(x)=3-4x g(x)=(3-x)/4

Ex. 3 Verify Inverses f(x)=2x3-1 g(x)= 3√(x+1)/2

To find the inverse of a function Replace f(x) with y Interchange x & y Does the new equation represent y as a function of x? If NO - f does not have an inverse If YES solve for y Replace y with f -1(x) Verify that f & f -1are inverses by showing that the domain of f is = to the range of f -1& range of f = domain of f -1

Ex. 5 Find the inverse of f(x)=(5-3x)/2 Step 1 y=(5-3x)/2 Step 2 x=(5-3y)/2 Step 3 yes, y is a function of x (solve for y) Step 4 f -1(x)=-2/3 x +5/3 Step 5 verify

Ex. Find f -1(x) for f(x)=5x-7 y=5x-7 x=5y-7 +7 +7 x+7=5y 5 5 y=(x+7)/5 Yes y is a function of x f -1(x)=(x+7)/5

Horizontal Line Test A function has an inverse if and only if no horizontal line intersects the graph of f at more than 1 point

Homework p. 243 1-4, 17-20 (just algebraically, 29-32, 33-37 odd, 39-59odd (don’t graph)