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Chapter 0: Problem 42

Use the vertical line test to determine whether the curve is the graph of a function of \(x\).

Short Answer

Expert verified
To determine if a curve is the graph of a function using the vertical line test, draw vertical lines at various points along the x-axis. If any vertical lines intersect the curve more than once, the curve is not the graph of a function. Otherwise, if no vertical line intersects the curve more than once, the curve is the graph of a function.

Step by step solution

01

Understanding the Vertical Line Test

To determine if a curve is a function using the vertical line test, you need to draw vertical lines at various points along the x-axis and see if they intersect the curve more than once. If they do, the graph represents a relation that is not a function. Remember that a function is defined as having a unique output (y-value) for each input (x-value). If a vertical line intersects the curve at multiple points, that means the same x-value would have multiple y-values, which is not a function.
02

Observe the Given Curve

Carefully observe the given curve and try to identify any points or regions where it seems like a vertical line could potentially intersect the graph more than once. Make note of these points and regions.
03

Draw Vertical Lines

Draw vertical lines through the notable points or regions you identified in Step 2, as well as other various points along the x-axis. The goal is to thoroughly test for any possible intersections occurring more than once.
04

Analyze the Intersections

Analyze the intersections of the vertical lines to see if any of them intersect the curve more than once. - If none of the vertical lines intersect the curve more than once, then the curve is the graph of a function. - If at least one vertical line intersects the curve more than once, then the curve is not the graph of a function. Once you determine the nature of the curve, you have completed the task.

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Most popular questions from this chapter

Let \(f(x)=x+\frac{1}{100} \sin 100 x\) a. Plot the graph of \(f\) using the viewing window \([-10,10] \times[-10,10]\) b. Plot the graph of \(f\) using the viewing window \([-0.1,0.1] \times[-0.1,0.1]\) c. Explain why the two displays obtained in parts (a) and (b) taken together give a complete description of the graph of \(f\).

a. If \(f(x)=x-1\) and \(h(x)=2 x+3\), find a function \(g\) such that \(h=g \circ f\). b. If \(g(x)=3 x+4\) and \(h(x)=4 x-8\), find a function \(f\) such that \(h=g \circ f\).

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=|x|, \quad y=2|x+1|-1\)

Suppose that \(f\) is a one-to-one function such that \(f(2)=5\). Find \(f^{-1}(5)\).

Find the zero(s) of the function f to five decimal places. $$ f(x)=2 x^{4}-4 x^{2}+1 $$
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