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**6.2 One-to-One Functions; Inverse Functions**

MAT SPRING 2009 6.2 One-to-One Functions; Inverse Functions In this section, we will study the following topics: One-to-one functions Finding inverse functions algebraically Finding inverse functions graphically Verifying that two functions are inverses, algebraically and graphically

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MAT SPRING 2009 Inverse Functions Example*: Solution:

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Inverse Functions Notice, for each of the compositions f(g(x)) and g(f(x)), the input was x and the output was x. That is because the functions f and g “undo” each other. In this example, the functions f and g are __________ ___________.

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**Definition of Inverse Functions**

MAT SPRING 2009 Definition of Inverse Functions The notation used to show the inverse of function f is f -1 (read “f-inverse”). Definition of Inverse Function f -1 is the inverse of function f f(f -1(x)) = x for every x in the domain of f -1 AND f -1(f(x))=x for every x in the domain of f The domain of f must be equal to the range of f -1, and the range of f must be equal to the domain of f -1. We will use this very definition to verify that two functions are inverses.

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**Watch out for confusing notation. **

MAT SPRING 2009 Watch out for confusing notation. Always consider how it is used within the context of the problem.

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MAT SPRING 2009 Example Use the definition of inverse functions to verify algebraically that are inverse functions.

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**A function f has an inverse if and only if f is ONE-TO-ONE.**

MAT SPRING 2009 One-to-one Functions Not all functions have inverses. Therefore, the first step in any of these problems is to decide whether the function has an inverse. A function f has an inverse if and only if f is ONE-TO-ONE. A function is one-to-one if each y-value is assigned to only one x-value.

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One-to-one Functions Using the graph, it is easy to tell if the function is one-to-one. A function is one-to-one if its graph passes the H____________________ L_________ T__________ ; i.e. each horizontal line intersects the graph at most ONCE. one-to-one NOT one-to-one

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Example Use the graph to determine which of the functions are one-to-one.

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**Finding the Inverse Function Graphically**

To find the inverse GRAPHICALLY: Make a table of values for function f. Plot these points and sketch the graph of f Make a table of values for f -1 by switching the x and y-coordinates of the ordered pair solutions for f Plot these points and sketch the graph of f -1 The graphs of inverse functions f and f -1 are reflections of one another in the line y = x.

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**Finding the Inverse Function Graphically (continued)**

Example: y=x f(x)=3x-5 x f(x) -5 1 -2 2 3 4 x f-1(x) -5 -2 1 2 4 3

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**Which of the following is the graph of the function below and its inverse?**

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**Finding the Inverse Function Algebraically**

To find the inverse of a function ALGEBRAICALLY: First, use the Horizontal Line Test to decide whether f has an inverse function. If f is not 1-1, do not bother with the next steps. Replace f(x) with y. Switch x and y Solve the equation for y. Replace y with f -1(x).

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**Finding the Inverse Function Algebraically**

Example Find the inverse of each of the following functions, if possible. 1.

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**Finding the Inverse Function Algebraically**

Example Find the inverse of each of the following functions, if possible. 2.

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**Finding the Inverse Function Algebraically**

Example (cont.) 3.

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**Finding the Inverse Function Algebraically**

Example (cont.) 4.

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Java Applet Click here to link to an applet demonstrating inverse functions.

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MAT SPRING 2009 End of Section 6.2

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