Question
What are the domain and range of the function n(x)=8(2/3)^x
Answers
GPT-4o mini
To find the domain and range of the function \( n(x) = 8 \left( \frac{2}{3} \right)^x \), we can analyze the function step by step.
### Domain
The domain of a function refers to all possible input values (x-values) for which the function is defined. In this case, the function \( n(x) \) is defined for all real numbers \( x \).
Thus, the domain is:
- **Domain**: \( (-\infty, \infty) \)
### Range
The range of a function refers to all possible output values (y-values) that the function can produce. The expression \( \left( \frac{2}{3} \right)^x \) is an exponential function that is decreasing because \( \frac{2}{3} < 1 \).
1. As \( x \) approaches \( -\infty \), \( \left( \frac{2}{3} \right)^x \) approaches \( +\infty \).
2. As \( x \) approaches \( +\infty \), \( \left( \frac{2}{3} \right)^x \) approaches \( 0 \) but never actually reaches \( 0 \).
Since the function is multiplied by \( 8 \), the function approaches \( 8 \cdot 0 = 0 \) as \( x \) goes to \( \infty \) but never actually reaches \( 0 \). Thus, the function can take any positive value up to but not including \( 0 \).
Therefore, the range is:
- **Range**: \( (0, +\infty) \)
### Summary
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (0, +\infty) \)
### Domain
The domain of a function refers to all possible input values (x-values) for which the function is defined. In this case, the function \( n(x) \) is defined for all real numbers \( x \).
Thus, the domain is:
- **Domain**: \( (-\infty, \infty) \)
### Range
The range of a function refers to all possible output values (y-values) that the function can produce. The expression \( \left( \frac{2}{3} \right)^x \) is an exponential function that is decreasing because \( \frac{2}{3} < 1 \).
1. As \( x \) approaches \( -\infty \), \( \left( \frac{2}{3} \right)^x \) approaches \( +\infty \).
2. As \( x \) approaches \( +\infty \), \( \left( \frac{2}{3} \right)^x \) approaches \( 0 \) but never actually reaches \( 0 \).
Since the function is multiplied by \( 8 \), the function approaches \( 8 \cdot 0 = 0 \) as \( x \) goes to \( \infty \) but never actually reaches \( 0 \). Thus, the function can take any positive value up to but not including \( 0 \).
Therefore, the range is:
- **Range**: \( (0, +\infty) \)
### Summary
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (0, +\infty) \)