Question

What are the domain and range of the function n(x) = 8(2/3) ^x?

A: Domain: all real numbers.
Range: n(x) > 0

B: Domain: x> 2/3
Range

C: Domain: all real numbers
Range : all real numbers

D: Domain : n(x) > 0
Range: x> 0

Answer 1

To analyze the function \( n(x) = 8 \left( \frac{2}{3} \right)^x \), we can determine its domain and range.

1. **Domain**: The base of the exponential function \( \left( \frac{2}{3} \right)^x \) is defined for all real numbers \( x \). Therefore, the domain of \( n(x) \) is all real numbers:
- Domain: all real numbers (or \( (-\infty, \infty) \)).

2. **Range**: Since \( \left( \frac{2}{3} \right) \) is a positive constant less than 1, the function \( n(x) \) represents an exponential decay. As \( x \) approaches infinity, \( n(x) \) approaches 0, but it never actually reaches 0. Thus, the range consists of positive values:
- Range: \( n(x) > 0 \).

Given this analysis, the correct option is:

**A: Domain: all real numbers.**
**Range: \( n(x) > 0 \)**.