Question
What are the domain and range of the function n(x)=8(2/3)x
Answers
GPT-4o mini
To determine the domain and range of the function \( n(x) = 8\left(\frac{2}{3}\right)^x \), we can analyze it step by step.
### Domain:
The function \( n(x) = 8\left(\frac{2}{3}\right)^x \) is an exponential function. Exponential functions are defined for all real numbers, so the domain of \( n(x) \) is:
\[
\text{Domain} = (-\infty, \infty)
\]
### Range:
To find the range, we look at the behavior of the function as \( x \) varies:
1. As \( x \) approaches infinity (\( x \to \infty \)), the term \( \left(\frac{2}{3}\right)^x \) approaches 0 because \( \frac{2}{3} < 1 \). Therefore, \( n(x) \) approaches 0.
2. As \( x \) approaches negative infinity (\( x \to -\infty \)), the term \( \left(\frac{2}{3}\right)^x \) approaches infinity because raising a fraction to a negative power results in a larger number. Thus, \( n(x) \) approaches infinity as well.
Since the function is always positive and approaches 0 but never touches it, the range of \( n(x) \) is:
\[
\text{Range} = (0, \infty)
\]
### Conclusion:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (0, \infty) \)
### Domain:
The function \( n(x) = 8\left(\frac{2}{3}\right)^x \) is an exponential function. Exponential functions are defined for all real numbers, so the domain of \( n(x) \) is:
\[
\text{Domain} = (-\infty, \infty)
\]
### Range:
To find the range, we look at the behavior of the function as \( x \) varies:
1. As \( x \) approaches infinity (\( x \to \infty \)), the term \( \left(\frac{2}{3}\right)^x \) approaches 0 because \( \frac{2}{3} < 1 \). Therefore, \( n(x) \) approaches 0.
2. As \( x \) approaches negative infinity (\( x \to -\infty \)), the term \( \left(\frac{2}{3}\right)^x \) approaches infinity because raising a fraction to a negative power results in a larger number. Thus, \( n(x) \) approaches infinity as well.
Since the function is always positive and approaches 0 but never touches it, the range of \( n(x) \) is:
\[
\text{Range} = (0, \infty)
\]
### Conclusion:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (0, \infty) \)