# Y axis y axis the only input which leads to an output

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Chapter P / Exercise 6
Trigonometry
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y -axis y -axis the only input which leads to an output of c y = c x -axis x -axis f ( x ) = mx + b linear functions f ( x ) = 1 constant functions any of these inputs leads to an output of 1 Figure 9.9: Does a horizontal line y = c intersect a curve once or more than once? 9.2.1 One-to-one Functions For a specified domain, one-to-one functions are functions with the prop- erty: Given any number c , there is at most one input x value in the do- main so that f ( x ) = c . Among our examples thus far, linear functions (degree 1 polynomials) are always one-to-one. However, f ( x ) = x 2 is not one-to-one; we’ve already seen that it can have two values for some of its inverses. By Fact 9.2.3, we can quickly come up with what’s called the horizontal line test. Important Fact 9.2.5 (Horizontal Line Test) . On a given domain of x -values, if the graph of some function f ( x ) has the property that every horizontal line crosses the graph at most only once, then the function is one-to-one on this domain.
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Chapter P / Exercise 6
Trigonometry
Larson Expert Verified
9.3. INVERSE FUNCTIONS 125 horizontal lines x-axis y-axis Figure 9.10: A one-to-one function f ( x ) = x 3 . Example 9.2.6. By the horizontal line test, it is easy to see that f ( x ) = x 3 is one-to-one on the domain of all real numbers. Although it isn’t common, it’s quite nice when a func- tion is one-to-one because we don’t need to worry as much about the number of input x values producing the same output y value. In effect, this is saying that we can define a “reverse process” for the function y = f ( x ) which will also be a function; this is the key theme of the next section. 9.3 Inverse Functions Let’s now come face to face with the problem of finding the “reverse pro- cess” for a given function y = f ( x ) . It is important to keep in mind that the domain and range of the function will both play an important role in this whole development. For example, Figure 9.11 shows the function f ( x ) = x 2 with three different domains specified and the corresponding range values. range range range domain domain domain Figure 9.11: Possible domains for a given range. These comments set the stage for a third important fact. Since the domain and range of the function and its inverse rule are going to be intimately related, we want to use notation that will highlight this fact. We have been using the letters x and y for the domain (input) and range (output) variables of f ( x ) and the “reverse process” is going to reverse these roles. It then seems natural to simply write y (instead of c ) for the input values of the “reverse process” and x for its output values.
126 CHAPTER 9. INVERSE FUNCTIONS Important Fact 9.3.1. Suppose a function f ( x ) is one-to-one on a domain of x values. Then define a NEW FUNCTION by the rule f - 1 ( y ) = the x value so that f ( x ) = y. The domain of y values for the function f - 1 ( y ) is equal to the range of the function f ( x ) .
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