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Chapter 1 | Functions and Graphs
Given an algebraic formula for a function f , the graph of f is the set of points ⎛ ⎝ x , f ( x ) ⎞ ⎠ , where x is in the domain of f and f ( x ) is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of f consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin. When creating a table of inputs and outputs, we typically check to determine whether zero is an output. Those values of x where f ( x ) =0 are called the zeros of a function . For example, the zeros of f ( x ) = x 2 −4 are x = ±2. The zeros determine where the graph of f intersects the x -axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the x -axis, or it may intersect multiple (or even infinitely many) times. Another point of interest is the y -intercept, if it exists. The y -intercept is given by ⎛ ⎝ 0, f (0) ⎞ ⎠ . Since a function has exactly one output for each input, the graph of a function can have, at most, one y -intercept. If x =0 is in the domain of a function f , then f has exactly one y -intercept. If x =0 is not in the domain of f , then f has no y -intercept. Similarly, for any real number c , if c is in the domain of f , there is exactly one output f ( c ), and the line x = c intersects the graph of f exactly once. On the other hand, if c is not in the domain of f , f ( c ) is not defined and the line x = c does not intersect the graph of f . This property is summarized in the vertical line test . Rule: Vertical Line Test Given a function f , every vertical line that may be drawn intersects the graph of f no more than once. If any vertical line intersects a set of points more than once, the set of points does not represent a function.
We can use this test to determine whether a set of plotted points represents the graph of a function ( Figure 1.8 ).
Figure 1.8 (a) The set of plotted points represents the graph of a function because every vertical line intersects the set of points, at most, once. (b) The set of plotted points does not represent the graph of a function because some vertical lines intersect the set of points more than once.
Example 1.3
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