Question
The range of an exponential function is \( (0, \infty) \). Select one: a. True b. False
Ask by Moreno Haynes. in Ghana
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The range of an exponential function is \( (0, \infty) \). So, the answer is **a. True**.
Solution
The exponential function is generally defined as
\[
f(x) = a^x,
\]
where \( a > 0 \) and \( a \neq 1 \).
1. The output of \( a^x \) is always positive, no matter what value \( x \) takes.
2. This means \( f(x) > 0 \) for all \( x \in \mathbb{R} \).
3. Therefore, the range of \( f(x) = a^x \) is \( (0, \infty) \).
Thus, the statement is:
a. True
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Simplify this solution Bonus Knowledge
Absolutely true! Exponential functions, like \(f(x) = a^x\) where \(a > 0\), always yield positive outputs regardless of whether \(x\) is positive, negative, or zero. This is due to the fact that they never reach zero or dip into negative values, meaning their range extends only above the x-axis, from 0 to infinity! To really get a grasp on exponential functions, consider how they model real-life scenarios like population growth or radioactive decay. For instance, the world population growth can be approximated by an exponential function. Understanding this application makes it clear just how powerful and ubiquitous exponential functions are in describing growth phenomena!
