# Inverse Functions

**Section 1.5 Inverse Functions
**

**Objective:**In this lesson you learned how to find inverses of functions

graphically and algebraically .

Important Vocabulary Define each term or
concept.Inverse function Let f and g be two functions . If f (g (x)) = x for every x in the domain of g and g (f (x)) = x for every x in the domain of f, then g is the inverse of the function f. The function g is denoted by f ^{-1}.One-to-one A function f is one-to-one if, for a and b in its domain, f (a) = f (b) implies a = b. Horizontal Line Test A function is one-to-one if every horizontal line intersects the graph of the function at most once. |

I. The Inverse of a Function (Pages 120-122)For a function f that is defined by a set of ordered pairs , to form the inverse function of f, . . . interchange the first and secondcoordinates of each of these ordered pairs. For a function f and its inverse f ^{-1}, the domain of f is equal tothe range of f , and the range of f is equal to^{-1}the domain of f .^{-1}To verify that two functions , f and g, are inverses of each other, . . . find f (g (x)) and g (f (x)). If both of these compositions are
equal to x for every x in the domain of the inner function, then the functions are inverses of each other. Example 1: Verify that the functions f (x) = 2x - 3 andare inverses of each other. |
What you should learn How to find inverse functions informally and verify that two functions are inverses of each other |

II. The Graph of an Inverse Function (Page 123)If the point (a, b) lies on the graph of f , then the point ( b , a ) lies on the graph of f ^{-1} and vice versa. Thegraph of f ^{-1 }is a reflection of the graph of f in the liney = x . |
What you should learn How to verify graphically and numerically that two functions are inverses of each other |

III. The Existence of an Inverse
Function (Page 124)A function f has an inverse f ^{-1} if and only if . . .f is one-to-one.If a function is one-to-one, that means . . . that no twoelements in the domain of the function correspond to the same element in the range of the function. To tell whether a function is one-to-one from its graph, . . . simply use the Horizontal Line Test, that is, check to see that
every horizontal line intersects the graph of the function at most once. Example 2: Does the graph of the function at the right have aninverse function? Explain. No, it doesn’t pass the Horizontal Line Test . |
What you should learn How to use graphs of functions to decide whether functions have inverses |

IV. Finding Inverse Functions Algebraically (Pages 125-126) To find the inverse of a function f algebraically , . . . 1) Use the Horizontal Line Test to decide whether f has aninverse. 2) In the equation for f (x), replace f (x) by y.3) Interchange the roles of x and y, and solve for y .
4) Replace y by f ^{-1}(x) in the new equation.5) Verify that f and f -1 are inverses of each other by showingthat f (f ^{-1}(x)) = x and f ^{-1}(f(x)) = x.Example 3: Find the inverse (if it exists) of f (x) = 4x - 5 .f ^{-1}(x) = 0.25x + 1.25 |
What you should learn How to find inverse functions algebraically |

Homework AssignmentPage(s) Exercises |

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