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Chapter 0: Problem 97

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=f(-x)\) for all \(x\) in the domain of \(f\), then the graph of \(f\) is symmetric with respect to the \(y\) -axis.

Short Answer

Expert verified
The statement is True. If \(f(x)=f(-x)\) for every \(x\) in its domain, then the function \(f\) is known as an even function, and its graph is symmetric with respect to the y-axis.

Step by step solution

01

Understand the Problem

In this exercise, the function \(f(x) = f(-x)\) is said to be symmetric around the y-axis. This means that if you were to draw the function out, any point on the left side of the y-axis has a corresponding point to the right of the y-axis, and vice versa.
02

Evaluate the Statement

Every function \(f\) that satisfies the property \(f(x) = f(-x)\) for all \(x\) in its domain is called an even function, and the graphs of even functions are indeed symmetric about the y-axis. Thus the statement is true and no counter-example is needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetry about the Y-Axis
Imagine folding a graph along the vertical y-axis, like closing a book. If the two halves match exactly, then the graph has symmetry about the y-axis. This happens when for every point with coordinates \( (x, y) \) on the graph, there exists a point with coordinates \( (-x, y) \) that is its mirror image across the y-axis.

Let's connect this to everyday life. Consider your face in a mirror; it's mostly symmetrical, right? If we think of your nose as the y-axis, each eye would be a point on opposite sides but at the same level, or 'height,' which in the graph is the y-value. This is how symmetry about the y-axis works in graphs of mathematical functions as well.
Properties of Even Functions
When we say a function is 'even', we aren't talking about even numbers, but rather a property of symmetry. Even functions are those satisfying the condition \( f(x) = f(-x) \) for all \( x \) in the function's domain. If you plug in a positive or negative value of \( x \) into an even function, you'll get the same result.

Some key aspects:
  • Their graphs are mirror images about the y-axis.
  • The function's highest power of \( x \) is even (like \( x^2 \), \( x^4 \), etc.).
  • The function's domain includes both positive and negative values of \( x \), and possibly zero.
Examples of even functions are \( f(x) = x^2 \) and \( f(x) = cos(x) \), where their graphs show the classic symmetry we've been discussing.
Evaluating Mathematical Statements
When faced with a mathematical statement, such as \( f(x) = f(-x) \) indicating symmetry about the y-axis, it's critical to distinguish between true and false claims. The process typically involves analyzing the definition and properties of the elements involved. In this case, for even functions.

Here's a simple checklist for evaluating whether such statements are true:
  • Understand the definition: Know what it means for a function to be even.
  • Apply what you know: Using sample values or algebraic proofs, check if the definition holds.
  • Consider counter-examples: If you suspect a statement might be false, look for exceptions that would prove it wrong.

In our textbook example, the claim was true since the property of even functions \( f(x) = f(-x) \) indeed results in y-axis symmetry.

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Most popular questions from this chapter

Automobile Costs The data in the table show the variable costs for operating an automobile in the United States for several recent years. The functions \(y_{1}, y_{2}\), and \(y_{3}\) represent the costs in cents per mile for gas and oil, maintenance, and tires, respectively. (Source: American Automobile Manufacturers Association) $$ \begin{array}{|c|c|c|c|} \hline \text { Year } & y_{1} & y_{2} & y_{3} \\ \hline 0 & 5.40 & 2.10 & 0.90 \\ \hline 1 & 6.70 & 2.20 & 0.90 \\ \hline 2 & 6.00 & 2.20 & 0.90 \\ \hline 3 & 6.00 & 2.40 & 0.90 \\ \hline 4 & 5.60 & 2.50 & 1.10 \\ \hline 5 & 6.00 & 2.60 & 1.40 \\ \hline 6 & 5.90 & 2.80 & 1.40 \\ \hline 7 & 6.60 & 2.80 & 1.40 \\ \hline \end{array} $$ (a) Use the regression capabilities of a graphing utility to find a cubic model for \(y_{1}\) and linear models for \(y_{2}\) and \(y_{3}\). (b) Use a graphing utility to graph \(y_{1}, y_{2}, y_{3}\), and \(y_{1}+y_{2}+y_{3}\) in the same viewing window. Use the model to estimate the total variable cost per mile in year \(12 .\)

Career Choice An employee has two options for positions in a large corporation. One position pays \(\$ 12.50\) per hour plus an additional unit rate of \(\$ 0.75\) per unit produced. The other pays \(\$ 9.20\) per hour plus a unit rate of \(\$ 1.30\). (a) Find linear equations for the hourly wages \(W\) in terms of \(x\), the number of units produced per hour, for each option. (b) Use a graphing utility to graph the linear equations and find the point of intersection. (c) Interpret the meaning of the point of intersection of the graphs in part (b). How would you use this information to select the correct option if the goal were to obtain the highest hourly wage?

Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) Write the volume \(V\) as a function of \(x\), the length of the corner squares. What is the domain of the function? (b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume. (c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.) $$ \begin{array}{|c|c|c|} \hline \text { Height, } x & \begin{array}{c} \text { Length } \\ \text { and Width } \end{array} & \text { Volume, } \boldsymbol{V} \\ \hline 1 & 24-2(1) & 1[24-2(1)]^{2}=484 \\ \hline 2 & 24-2(2) & 2[24-2(2)]^{2}=800 \\ \hline \end{array} $$

Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$f(x)=\frac{1}{2} x^{3}+2$$

Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=3 x-1\) \(\frac{f(x)-f(1)}{x-1}\)
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