Question
What is the range of the given exponential function below? \( 3^{x}+2 \) \( (-\infty, \infty) \) \( (2, \infty) \) \( [2, \infty) \) \( (1, \infty) \)
Ask by Barnett Beck. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The range of the function \( 3^{x} + 2 \) is \( (2, \infty) \).
Solution
To determine the range of the exponential function \( f(x) = 3^x + 2 \), let's analyze its behavior:
1. **Basic Exponential Function:** The function \( 3^x \) is always positive for all real numbers \( x \). Its range is \( (0, \infty) \).
2. **Transformation:** Adding 2 to \( 3^x \) shifts the entire graph upward by 2 units. Therefore, the new function \( 3^x + 2 \) will always be greater than 2.
3. **Limits:**
- As \( x \) approaches \( -\infty \), \( 3^x \) approaches 0, so \( 3^x + 2 \) approaches 2 but never actually reaches it.
- As \( x \) approaches \( \infty \), \( 3^x \) grows without bound, so \( 3^x + 2 \) also grows without bound.
Combining these observations, the range of \( f(x) = 3^x + 2 \) is all real numbers greater than 2.
**Correct Answer:** \( (2, \infty) \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
The exponential function \( 3^{x} \) always produces positive values, ranging from 0 to infinity as \( x \) changes. When you add 2 to this function, it shifts the entire output up by 2 units, meaning the lowest value it can take is 2 when \( x \) approaches negative infinity. This results in a range starting at 2 and extending to infinity. As a result, the range of the function \( 3^{x} + 2 \) is expressed as \([2, \infty)\). Isn’t it fascinating how just a little tweak can change everything? You can visualize this as raising the entire graph of \( 3^{x} \) upward!
