Question

what are the domain and rangeof the function n(x) = 8(2/3)^x

Answer 1

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

### Domain:
The function \( n(x) \) is defined for all real numbers \( x \) because there are no restrictions on \( x \) when evaluating the expression. Therefore, the domain of \( n(x) \) is:
\[
\text{Domain: } (-\infty, \infty)
\]

### Range:
The function \( n(x) \) is an exponential function, where the base \( \frac{2}{3} \) is between 0 and 1. As \( x \) approaches \( \infty \), \( \left(\frac{2}{3}\right)^x \) approaches 0, making \( n(x) \) approach 0 as well. As \( x \) approaches \( -\infty \), \( \left(\frac{2}{3}\right)^x \) grows larger without bound. Since \( n(x) \) is always positive (as it is multiplied by 8, a positive constant), the function never actually reaches 0.

Thus, the range of the function is:
\[
\text{Range: } (0, \infty)
\]

### Summary:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (0, \infty) \)