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Chapter 3: Problem 99

Can a function be both even and odd? Explain.

Short Answer

Expert verified
Yes, a function can be both even and odd. The only function that satisfies both conditions is the zero function.

Step by step solution

01

Define Even and Odd Functions

A function is even if \[ f(-x) = f(x) \] for all values of x. A function is odd if \[ f(-x) = -f(x) \] for all values of x.
02

Examine if a Function Can Satisfy Both Conditions

To determine if a function can be both even and odd, we need to find whether there exists any function that satisfies both \[ f(-x) = f(x) \] and \[ f(-x) = -f(x) \] simultaneously.
03

Equate the Two Conditions

If a function is both even and odd, then it must hold that \[ f(-x) = f(x) = -f(x). \]
04

Solve the Equation

From \[ f(x) = -f(x), \] we add \[ f(x) \] to both sides to get \[ 2f(x) = 0. \] Therefore, \[ f(x) = 0. \] This implies that the only function that can be both even and odd is the zero function.
05

Conclusion

We conclude that the only function that can be both even and odd is the zero function, denoted as \[ f(x) = 0. \]

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Key Concepts

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

zero function
The zero function is an interesting concept in mathematics. It is defined as \[ f(x) = 0 \] for all values of x. This means that regardless of the input, the output is always zero. One of the unique properties of the zero function is its ability to be both even and odd. Since \[ f(-x) = f(x) = 0 \] and \[ f(-x) = -f(x) = 0, \] it meets the criteria for both even and odd functions. The zero function is a constant function, which simplifies many mathematical processes and serves as a fundamental building block in various areas of math.
symmetry in functions
Symmetry in functions is an important topic that can help understand and analyze different types of functions. If a function is even, it will have symmetry about the y-axis. This means that the graph on the left side of the y-axis is a mirror image of the graph on the right side. Mathematically, this can be represented as \[ f(-x) = f(x). \] On the other hand, a function is said to be odd if it displays origin symmetry. This means the graph of the function is symmetric with respect to the origin, and flipping the graph over both the x-axis and y-axis gives the same graph. This is represented by \[ f(-x) = -f(x). \] Understanding these symmetries can simplify solving equations and analyzing the behavior of functions.
function properties
Functions have various properties that can help determine their nature and behavior. One key property is whether a function is even, odd, or neither. This is determined by the function's response to negative inputs. Another important property is continuity, which means the function has no breaks or jumps in its graph. Differentiability is another property, which implies that the function's derivative exists at every point in its domain. Some functions, like the zero function, have the unique property of being both even and odd. Understanding these properties can aid in solving complex mathematical problems and give insights into the function's characteristics.

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Most popular questions from this chapter

Use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. \(f(x)=-0.4 x^{4}-0.5 x^{3}+0.8 x^{2}-2 \quad[-3,2]\)

Challenge Problem If \(f\left(\frac{x+4}{5 x-4}\right)=3 x^{2}-2,\) find \(f(1)\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Angie runs 7 mph for the first half of a marathon, 13.1 miles, but twists her knee and must walk 2 mph for the second half. What was her average speed? Round to 2 decimal places.

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=x^{2}+2 x\)

Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: (0,-3) and (3,0)\(;\) a local maximum value of -2 at \(-1 ;\) a local minimum value of -6 at \(2 .\) Compare your graph with those of others. Comment on any differences.
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