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Chapter 1 | Functions and Graphs
1.4 | Inverse Functions
Learning Objectives 1.4.1 Determine the conditions for when a function has an inverse. 1.4.2 Use the horizontal line test to recognize when a function is one-to-one.
1.4.3 Find the inverse of a given function. 1.4.4 Draw the graph of an inverse function. 1.4.5 Evaluate inverse trigonometric functions.
An inverse function reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it. In this section, we define an inverse function formally and state the necessary conditions for an inverse function to exist. We examine how to find an inverse function and study the relationship between the graph of a function and the graph of its inverse. Then we apply these ideas to define and discuss properties of the inverse trigonometric functions. Existence of an Inverse Function We begin with an example. Given a function f and an output y = f ( x ), we are often interested in finding what value or values x were mapped to y by f . For example, consider the function f ( x ) = x 3 +4. Since any output y = x 3 +4, we can solve this equation for x to find that the input is x = y −4 3 . This equation defines x as a function of y . Denoting this function as f −1 , and writing x = f −1 ( y ) = y −4 3 , we see that for any x in the domain of f , f −1 ⎛ ⎝ f ( x ) ⎞ ⎠ = f −1 ⎛ ⎝ x 3 +4 ⎞ ⎠ = x . Thus, this new function, f −1 , “undid” what the original function f did. A function with this property is called the inverse function of the original function. Definition Given a function f with domain D and range R , its inverse function (if it exists) is the function f −1 with domain R and range D such that f −1 ( y ) = x if f ( x ) = y . In other words, for a function f and its inverse f −1 , (1.11) f −1 ⎛ ⎝ f ( x ) ⎞ ⎠ = x for all x in D , and f ⎛ ⎝ f −1 ( y ) ⎞ ⎠ = y for all y in R .
Note that f −1 is read as “f inverse.” Here, the −1 is not used as an exponent and f −1 ( x ) ≠1/ f ( x ). Figure 1.37 shows the relationship between the domain and range of f and the domain and range of f −1 .
Figure 1.37 Given a function f and its inverse f −1 , f −1 ( y ) = x if and only if f ( x ) = y . The range of f becomes the domain of f −1 and the domain of f becomes the range of f −1 .
Recall that a function has exactly one output for each input. Therefore, to define an inverse function, we need to map each
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