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Chapter 1: Problem 78

Can a function be both even and odd? Give reasons for your answer.

Short Answer

Expert verified
A function can be both even and odd only if it is the zero function.

Step by step solution

01

Understand the Definitions

An even function satisfies the condition \(f(x) = f(-x)\) for all values in its domain. An odd function satisfies \(f(-x) = -f(x)\) for its domain. These definitions will guide us in determining if a function can be both even and odd.
02

Analyze Simultaneous Equations

If a function is both even and odd, then it must satisfy both conditions: \(f(x) = f(-x)\) and \(f(-x) = -f(x)\). Setting these equations equal due to the same argument \(f(-x)\), equate \(f(-x) = f(x)\) to \(f(-x) = -f(x)\).
03

Solve the Equations

From the two conditions, equating \(f(x) = -f(x)\), we find that \(2f(x) = 0\). This simplifies to \(f(x) = 0\), meaning the only function that satisfies both conditions is \(f(x) = 0\) for all \(x\).
04

Conclude with Explanation

Since the only function that can be both even and odd is the trivial zero function, no non-zero function can be both even and odd. Therefore, in general, a function cannot be both even and odd unless it is identically zero.

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Key Concepts

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

Function Properties
Functions have distinct properties that help to classify and understand their behavior. These properties are important to grasp as they dictate how a function responds to inputs within its domain.

To categorize functions, we look at symmetries. Namely, even and odd functions possess unique symmetries:
  • Even Function: Symmetric with respect to the y-axis. The graph of the function will look the same when you fold it along the y-axis.
  • Odd Function: Symmetric with respect to the origin. If you rotate the graph 180 degrees around the origin, the graph appears unchanged.
Understanding these properties helps predict and calculate function behavior efficiently.
Even Function Definition
An even function has a special characteristic, captured by the equation \( f(x) = f(-x) \).

This means for every input 'x,' the output of the function will be the same as the output for '-x.'

Graphically, this reflects as symmetry along the y-axis. Common examples of even functions include \( f(x) = x^2 \) and \( f(x) = \cos x \). These functions will display a mirror-like symmetry, providing a helpful way to visualize and verify their even nature in practical scenarios.
Odd Function Definition
Odd functions, on the other hand, adhere to the condition \( f(-x) = -f(x) \).

This implies that flipping the sign of your input results in the negation of the output. So if you substitute '-x' into the function, you should achieve the negative of what you get with 'x.'

In terms of visualization, odd functions exhibit rotational symmetry about the origin. Examples of odd functions include \( f(x) = x^3 \) and \( f(x) = \sin x \). Testing these equations can solidify understanding and assist in identifying when functions exhibit these characteristics.

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Most popular questions from this chapter

Use graphing software to graph the functions specified.Select a viewing window that reveals the key features of the function. Graph the upper branch of the hyperbola \(y^{2}-16 x^{2}=1\)

Graph the functions in Exercises \(35-54\) $$ y=|x-2| $$

Use graphing software to graph the functions specified.Select a viewing window that reveals the key features of the function. Graph the function \(f(x)=\sin ^{3} x\)

Exercises \(25-34\) tell how many units and in what directions the graphs of the given equations are shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ x^{2}+y^{2}=25 \quad \text { Up } 3, \text { left } 4 $$

In Exercises 17 and \(18,\) (a) write formulas for \(f \circ g\) and \(g \circ f\) and find the \((\mathbf{b})\) domain and \((\mathbf{c})\) range of each. $$ f(x)=x^{2}, g(x)=1-\sqrt{x} $$
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