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Chapter 3: Problem 83

Is the function \(f\) defined by \(f(x)=2^{x}\) for every real number \(x\) an even function, an odd function, or neither?

Short Answer

Expert verified
The function \(f(x) = 2^x\) is neither even nor odd because it does not satisfy the conditions \(f(x) = f(-x)\) for even functions and \(f(-x) = -f(x)\) for odd functions.

Step by step solution

01

Write down the given function

We are given the function \(f(x) = 2^x\) for every real number \(x\).
02

Calculate f(-x)

To check if the function is even or odd, we need to evaluate \(f(-x)\). \(f(-x) = 2^{-x}\)
03

Compare f(-x) to f(x) and -f(x)

Now, we compare \(f(-x)\) to \(f(x)\) and \(-f(x)\). - For even function, \(f(x) = f(-x)\). Check if there is any equivalence between the functions: \(f(x) = 2^x\) \(f(-x) = 2^{-x}\) As \(2^x\) is not equal to \(2^{-x}\), the function is not even. - For an odd function, \(f(-x) = -f(x)\). Check if there is any equivalence between the functions: \(-f(x) = -(2^x)\) \(f(-x) = 2^{-x}\) As \(- (2^x)\) is not equal to \(2^{-x}\), the function is not odd.
04

Determine if the function is even, odd, or neither

The function satisfies neither the conditions of even functions nor the conditions of odd functions. Therefore, the function is neither even nor odd.

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Most popular questions from this chapter

Suppose \(b\) is a small positive number. Estimate the slope of the line containing the points \(\left(e^{3}, 5+b\right)\) and \(\left(e^{3+b}, 5\right)\).

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