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: \( (0, \infty) \)
3 answers