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Chapter 8: Problem 60

Explain how the vertical line test is used to determine whether a graph represents a function.

Short Answer

Expert verified
The vertical line test is a method to determine whether a graph represents a function. It states that if a vertical line can be drawn anywhere on the graph and intersect the graph in only one place, the graph represents a function. If the vertical line intersects the graph at more than one point, the graph does not represent a function.

Step by step solution

01

Definition of a Function

Before applying the vertical line test, it is important to understand what a function is. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Therefore, in the context of a graph, no single input (x-coordinate) can map to multiple outputs (y-coordinates).
02

Understanding the Vertical Line Test

The vertical line test is used to determine whether a relation is a function or not. The vertical line test states that a graph represents a function if and only if a vertical line drawn at any place on the graph intersects the graph at most one point.
03

Application of the Vertical Line Test

To apply the vertical line test, imagine placing a vertical line anywhere on the x-axis of the graph. If, at any point, the vertical line intersects the graph at more than one point, the graph does not represent a function. On the other hand, if the vertical line only ever intersects the graph at one point regardless of where it is placed, the graph represents a function.

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Most popular questions from this chapter

The regular price of a computer is \(x\) dollars. Let \(f(x)=x-400\) and \(g(x)=0.75 x\) a. Describe what the functions \(f\) and \(g\) model in terms of the price of the computer. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the computer. c. Repeatpart(b) for \((g \circ f)(x)\). d. Which composite function models the greater discount on the computer, \(f \circ g\) or \(g \circ f ?\) Explain. e. Find \(f^{-1}\) and describe what this models in terms of the price of the computer.

Find a. \((f \circ g)(x)\), b. \((g \circ f)(x)\), c. \((f \circ g)(2)\). $$f(x)=x^{2}+2, \quad g(x)=x^{2}-2$$

Graph each of the three functions in the same \([-10,10,1]\) by \([-10,10,1]\) viewing rectangle. \(y_{1}=x-4\) \(y_{2}=2 x\) \(y_{3}=y_{1}-y_{2}\)

\(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{c|c}\hline x & f(x) \\\\\hline-1 & 1 \\\\\hline 0 & 4 \\\\\hline 1 & 5 \\\\\hline 2 & -1 \\ \hline\end{array}$$ $$\begin{array}{c|c}\hline x & g(x) \\\\\hline-1 & 0 \\\\\hline 1 & 1 \\\\\hline 4 & 2 \\\\\hline 10 & -1 \\ \hline\end{array}$$ $$(g \circ f)(-1)$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and g are inverses. $$f(x)=\frac{1}{x}+2, \quad g(x)=\frac{1}{x-2}$$
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