Answer :
To determine the domain and range of the function \( f(x) = \left( \frac{1}{5} \right)^x \), let's analyze the function step-by-step.
### Step 1: Determine the Domain
The domain of a function represents all the possible input values (\( x \)) for which the function is defined. For the exponential function \( f(x) = a^x \), where \( a > 0 \) and \( a \neq 1 \), the base \( a \) is a positive real number. This means that you can substitute any real number for \( x \), and the function will produce a valid output.
Since \( \left( \frac{1}{5} \right) \) is a positive number and any real number can be an exponent, the domain of \( f(x) = \left( \frac{1}{5} \right)^x \) is:
[tex]\[ \text{Domain: All real numbers} \][/tex]
### Step 2: Determine the Range
The range of a function represents all the possible output values (\( f(x) \)) that the function can produce. For the function \( f(x) = \left( \frac{1}{5} \right)^x \):
- When \( x = 0 \), \( f(x) = \left( \frac{1}{5} \right)^0 = 1 \).
- As \( x \) becomes larger and larger (going towards \( +\infty \)), \( f(x) \) approaches 0 but is never actually 0. It asymptotically approaches 0 from the positive side.
- As \( x \) becomes more and more negative (going towards \( -\infty \)), \( f(x) = \left( \frac{1}{5} \right)^x = 5^{-x} \) becomes very large.
Since \( \left( \frac{1}{5} \right)^x \) is always positive for any real number \( x \), the range of the function is:
[tex]\[ \text{Range: All real numbers greater than zero} \][/tex]
### Conclusion
Combining the domain and range, we have:
- Domain: All real numbers
- Range: All real numbers greater than zero
Therefore, the correct answer is:
[tex]\[ \text{The domain is all real numbers. The range is all real numbers greater than zero.} \][/tex]
### Step 1: Determine the Domain
The domain of a function represents all the possible input values (\( x \)) for which the function is defined. For the exponential function \( f(x) = a^x \), where \( a > 0 \) and \( a \neq 1 \), the base \( a \) is a positive real number. This means that you can substitute any real number for \( x \), and the function will produce a valid output.
Since \( \left( \frac{1}{5} \right) \) is a positive number and any real number can be an exponent, the domain of \( f(x) = \left( \frac{1}{5} \right)^x \) is:
[tex]\[ \text{Domain: All real numbers} \][/tex]
### Step 2: Determine the Range
The range of a function represents all the possible output values (\( f(x) \)) that the function can produce. For the function \( f(x) = \left( \frac{1}{5} \right)^x \):
- When \( x = 0 \), \( f(x) = \left( \frac{1}{5} \right)^0 = 1 \).
- As \( x \) becomes larger and larger (going towards \( +\infty \)), \( f(x) \) approaches 0 but is never actually 0. It asymptotically approaches 0 from the positive side.
- As \( x \) becomes more and more negative (going towards \( -\infty \)), \( f(x) = \left( \frac{1}{5} \right)^x = 5^{-x} \) becomes very large.
Since \( \left( \frac{1}{5} \right)^x \) is always positive for any real number \( x \), the range of the function is:
[tex]\[ \text{Range: All real numbers greater than zero} \][/tex]
### Conclusion
Combining the domain and range, we have:
- Domain: All real numbers
- Range: All real numbers greater than zero
Therefore, the correct answer is:
[tex]\[ \text{The domain is all real numbers. The range is all real numbers greater than zero.} \][/tex]
