# Inverse Functions

Overview • Section 4.1 in the textbook: – Introduction to Inverse Functions – Graphing Inverse Functions – Composition of a Function and Its Inverse – Finding Inverse Functions |

Introduction to Inverse Functions |

Inverses & One-to-one Functions • Recall that for each point in a function: – Each x- coordinate is associated with only ONEy-coordinate • Given a function f consisting of a set of points,let f denote the relation that results if we swap^{ -1}the x and y coordinates of each point in f• If f^{ -1} is also a function, then it is called theinverse of f AND f is classified as a one-to- functionone – Ex: f = {(0, 0), (1, 5), (-3, 4)}→ f = {(0, 0), (5, 1),^{-1}(4, -3)} → f is a one-to-one function |

• The domain and range of a function and its inverse are switched:– Ex: f = {(0, 0), (1, 5), (-3, 4)} →f^{-1} = {(0, 0), (5, 1), (4, -3)} →f (domain) = {-3, 0, 1} & f (range) = {0, 4, 5} →f^{-1} (domain) = {0, 4, 5} & f^{-1} (range) = {-3, 0, 1}• To test a graph for an inverse, we use the horizontal line test – If the horizontal line crosses the graph more than once, the graph does not have an inverse |

Inverses & One-to-one Functions (Example) Ex 1: Determine whether each function hasan inverse. If the function does not have an inverse, state why: a) {(0, 0), (1, 1), (4, 2), (9, 3), (16, 4)} b) {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)} |

Ex 2: Determine which graphs have aninverse: |

Graphing Inverse Functions |

Graphing Inverses • Given the equation of f AND that f is aone-to-one function, we can sketch agraph of f^{-1} – Swap the x and y coordinates of f• The graph of f and its inverse f^{-1} issymmetric to the line y = x – If we folded a piece of paper over the line y = x, f and f^{-1} would lie on top of each other |

Graphing Inverses (Example)Ex 3: Given the graph of f (x), sketch f^{ -1} (x),the inverse of f (x): |

Composition of a Function and Its Inverse |

Composition of a Function and Its Inverse • Suppose we start off with the value 8 – If we multiply 8 by 2, we get 16 – If we divide 16 by 2, we get back to the original 8 – Works the same way if we divide first and then multiply • Multiplication & division are inverse operations– One undoes the effect of the other • Recall that a composition takes the output ofone function as the input to another – In our case, the output of the first “function” was a product which was then put into a division “function” |

• Given a one-to-one function f, f^{-1} is theinverse of f if and only if– (f ◦ f )(x) = x^{ -1}• The composition of f and f^{-1} yields the originalvalue • AND– (f (x) = x^{ -1} ◦ f)• The composition of f and^{ -1} f yields the originalvalue |

Composition of a Function and Its Inverse (Example) Ex 4: Determine whether the given functions fand g are inverses: |

Finding the Inverse of a Function |

Finding the Inverse of a Function • Rather than a graph of f^{-1} , we more often thannot want the equation of f^{-1} • Recall that to find f^{-1} when f consisted of a set ofpoints, we switched the x and y coordinates • Given that f is a one-to-one function, to find f^{-1}
:– Replace f (x) with y– Switch x and y– Solve for y – Substitute f^{-1} (x) for y• Remember how to check whether two functions are inverses |

Given that the function is one-to-one, findEx 5: its inverse: |

Summary • After studying these slides, you should know how to do the following: – Understand the concept of an inverse function – State whether a function has an inverse by looking at its graph – State the domain and range of a function and its inverse – Graph the inverse of a function – State whether two functions are inverses by using composition – Find the inverse of a function • Additional Practice – See the list of suggested problems for 4.1 • Next lesson – Exponential Functions (Section 4.2) |

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