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Chapter 13: Problem 9

For an exponential function of the form \(f(x)=a^{x}\) \((a>0, a \neq 1),\) answer the following. What is the domain?

Short Answer

Expert verified
The domain of the function \(f(x) = a^x\) is \(\{x \in \mathbb{R}\}\), which means it includes all real numbers.

Step by step solution

01

Identify the given function

The given function is \(f(x) = a^x\), where \(a > 0\) and \(a \neq 1\).
02

Determine the input values (x-values) for which the function is defined

Recall that the base, \(a\), of an exponential function must be positive and not equal to one. This means that the function is always defined, since raising a positive number to any power always results in a valid, finite result. Therefore, the function \(f(x) = a^x\) is defined for all real numbers x.
03

Write the domain of the function

Since the function is defined for all real numbers x, the domain of \(f(x) = a^x\) is: Domain (D): \(\{x \in \mathbb{R}\}\) This means that the domain includes all real numbers.

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