Prerequisites Section P 4 Compositions and Inverses Copyright

Prerequisites Section P. 4 Compositions and Inverses Copyright © 2011 Pearson Education, Inc.

P. 4 Compositions of Functions If f and g are two functions, the composition of f and g, written f ◦ g, is a function that is defined by the equation (f ◦ g)(x) = f(g(x)), provided that g(x) is in the domain of f. The composition of g and f is written g ◦ f. Copyright © 2011 Pearson Education, Inc. 3

P. 4 One-to-One Functions If a function has no two ordered pairs with different first coordinates and the same coordinate, then the function is a one-to-one function. Copyright © 2011 Pearson Education, Inc. 4

P. 4 Invertible Functions The inverse of a one-to-one function f is the function f– 1 (read “f inverse”), where the ordered pairs of f– 1 are obtained by interchanging the coordinates in each ordered pair of f. Copyright © 2011 Pearson Education, Inc. 5

P. 4 Inverse Functions To find the inverse of an invertible function given in function notation: 1. Replace f(x) by y. 2. Interchange x and y. 3. Solve the equation for y. 4. Replace y by f– 1(x). Copyright © 2011 Pearson Education, Inc. 6

P. 4 Graphs of f and f– 1 If a point (a, b) is on the graph of an invertible function f, then (b, a) is on the graph of f– 1. The graph of f– 1 is a reflection of the graph of f with respect to the line y = x. Copyright © 2011 Pearson Education, Inc. 7

P. 4 Graphs of f and f– 1 Figure P. 46 Copyright © 2011 Pearson Education, Inc. 8