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Chapter 3: Problem 60

(a) A function \(f\) is said to be even if the equation \(f(-x)=f(x)\) is satisfied by all values of \(x\) in the domain of \(f .\) Explain why the graph of an even function must be symmetric about the \(y\) -axis. (b) Show that each function is even by computing \(f(-x)\) and then noting that \(f(x)\) and \(f(-x)\) are equal. (i) \(f(x)=x^{2}\) (iii) \(f(x)=3 x^{6}-\frac{4}{x^{2}}+1\) (ii) \(f(x)=2 x^{4}-6\)

Short Answer

Expert verified
Even functions satisfy \(f(-x) = f(x)\), leading to symmetry about the \(y\)-axis. All given functions are even.

Step by step solution

01

Understanding Even Functions

A function is considered even if for every value of \(x\) in its domain, \(f(-x) = f(x)\). This definition implies that for every point \(x, f(x)\) on the graph, there is a corresponding point \(-x, f(-x)\). Since \(f(-x) = f(x)\), both points will have the same \(y\)-coordinate, forming a mirror image across the \(y\)-axis, demonstrating symmetry about the \(y\)-axis.
02

Check Evenness for \(f(x)=x^2\)

Compute \(f(-x)\): \(f(-x) = (-x)^2 = x^2\). Since \(f(x) = x^2\) and \(f(-x) = x^2\), \(f(-x) = f(x)\). Thus, \(f(x) = x^2\) is an even function.
03

Check Evenness for \(f(x)=3x^6-\frac{4}{x^2}+1\)

Compute \(f(-x)\): \(f(-x) = 3(-x)^6 - \frac{4}{(-x)^2} + 1 = 3x^6 - \frac{4}{x^2} + 1\), which simplifies to the original function \(f(x) = 3x^6 - \frac{4}{x^2} + 1\). Since \(f(-x) = f(x)\), this function is even.
04

Check Evenness for \(f(x)=2x^4-6\)

Compute \(f(-x)\): \(f(-x) = 2(-x)^4 - 6 = 2x^4 - 6\). Since \(f(x) = 2x^4 - 6\), this confirms that \(f(-x) = f(x)\), making \(f(x)\) even.

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Key Concepts

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

Function Symmetry
Function symmetry is an interesting concept that describes how certain functions can look the same when viewed from different angles or positions. If a function is symmetric, this means its graph has a mirror-like quality.
Symmetry can happen around different axes, such as the x-axis, y-axis, or even the origin. For even functions, we specifically focus on symmetry around the y-axis. Symmetry around an axis means that for every point on one side of the axis, there is another point on the opposite side at the same distance. This makes the graph a perfect mirror image from one side to the other.
  • For example, the parabola shaped graph of the basic function like \(f(x)=x^2\) is symmetric about the y-axis.
  • If you fold the graph along the y-axis, both halves of the graph would line up perfectly.
This particular symmetry is crucial for identifying even functions and helps greatly in understanding other concepts in mathematics.
Y-Axis Symmetry
Y-axis symmetry is a key characteristic of even functions. It means that each part of the graph on one side of the y-axis mirrors the opposite side. Imagine drawing a vertical line along the y-axis.
If the graph folds evenly over this line, then it has y-axis symmetry. To determine symmetry, we check if \(f(-x) = f(x) \) for all x in the domain of the function. This means replacing x with -x should result in the same function value, confirming the mirror effect.
  • For instance, with \(f(x) = x^2\), when calculated as \(f(-x) = (-x)^2 = x^2\), we see it matches the original function \(f(x)\).
  • Thus, confirming y-axis symmetry.
Additionally, for a function like \(f(x) = 3x^6 - \frac{4}{x^2} + 1\), calculating \(f(-x)\) shows us \(f(-x) = 3x^6 - \frac{4}{x^2} + 1\) which is equal to the original \(f(x)\). This analysis proves the function is even with y-axis symmetry.
Polynomial Functions
Polynomial functions are a broad category of functions, which include terms in the form of constants multiplied by variables raised to whole number powers. They can be simple like \(f(x) = x^2\), or more complex like \(f(x) = 3x^6 - \frac{4}{x^2} + 1\). An understanding of polynomial functions helps when determining symmetry.
Symmetry, especially y-axis symmetry, can easily be checked in polynomials. For even polynomial functions, all the terms have even powers or are constants.
  • For example, in \(f(x) = 2x^4 - 6\), each term is an even power or constant. Hence, it is an even function, showing y-axis symmetry.
  • Complex polynomials like \(f(x) = 3x^6 - \frac{4}{x^2} + 1\) can also be checked for symmetry using the same principle.
Recognizing and understanding polynomial structures help to quickly identify the symmetry without full calculations, making analysis more efficient.

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Most popular questions from this chapter

Use a graphing utility to graph each function and then apply the horizontal line test to see whether the function is one-to-one. (a) \(f(x)=x^{3}+x^{2}+x\) (b) \(g(x)=x^{3}-x^{2}+x\) (c) \(h(x)=x^{3}-x^{2}-x\)

Suppose that in a certain biology lab experiment, the number of bacteria is related to the temperature \(T\) of the environment by the function $$N(T)=-2 T^{2}+240 T-5400 \quad(40 \leq T \leq 90)$$ Here, \(N(T)\) represents the number of bacteria present when the temperature is \(T\) degrees Fahrenheit. Also, suppose that \(t\) he experiment begins, the temperature is given by $$T(t)=10 t+40 \quad(0 \leq t \leq 5)$$ (a) Compute \(N[T(t)]\) (b) How many bacteria are present when \(t=0\) hr? When \(t=2 \mathrm{hr} ?\) When \(t=5 \mathrm{hr} ?\)

If \(g(x)=x^{2}-3 x k-4\) and \(g(1)=-2,\) find \(k\)

A function \(f\) is given. Say how the graph of each of the related functions can be obtained from the graph of \(f\), and then use a graphing utility to verify your statement (as in Figure 11 ). \(f(x)=-x^{3}+3 x^{2}-3 x+1\) (a) \(y=-x^{3}+3 x^{2}-3 x-1\) (b) \(y=x^{3}+3 x^{2}+3 x+1\) (c) \(y=x^{3}-3 x^{2}+3 x-1\)

\(\pi=3.141592653589793 \ldots\) and so on! For each natural number \(n,\) let \(G(n)\) be the digit in the \(n\) th decimal place of \(\pi .\) For instance, according to the expression for \(\pi\) given above, we have \(G(1)=1, G(2)=4,\) and \(G(5)=9\) (a) Use the expression for \(\pi\) given above to evaluate \(G(10)\) and \(G(14)\) (b) Use the Internet to help you evaluate \(G(100), G(750)\) and \(G(1000) .\) Suggestions: Using any of the common search engines on the World Wide Web, under the categories of mathematics or science, search for "pi." Here, for example, are two sites that contain the information you need. (They were accessible at the time of this writing, March 2004 .)
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