To find the domain of the function $y = \left(\frac{1}{5}\right)^x$, let's go through the process step-by-step.

Step 1: Understanding the Function

The given function is an exponential function of the form:

$y = a^x$

where $a = \frac{1}{5}$. For exponential functions like this, $a$ is a constant positive number, and the variable $x$ is the exponent.

Step 2: Analyzing the Domain of Exponential Functions

In general, the domain of an exponential function $a^x$, where $a > 0$ and $a \neq 1$, is all real numbers. This is because for any real value of $x$, the expression $a^x$ will always be well-defined. There are no restrictions on $x$ (such as division by zero or taking even roots of negative numbers).

Step 3: Applying This to the Given Function

For $y = \left(\frac{1}{5}\right)^x$, the base $\frac{1}{5}$ is a positive number, and there are no restrictions on the value of $x$. Therefore, the function is defined for all real values of $x$.

Step 4: Conclusion

Thus, the domain of the function $y = \left(\frac{1}{5}\right)^x$ is:

$\boxed{(-\infty, \infty)}$

Step 5: Explanation of Domain

• The domain refers to the set of all possible input values $x$ for which the function is defined.
• In the case of exponential functions with positive bases, there are no inherent limitations on the input values, meaning $x$ can take any real number.

Would you like further details or examples on domains of other types of functions?

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Related Questions:

1. What is the range of the function $y = \left(\frac{1}{5}\right)^x$?
2. How do you find the domain of a logarithmic function, such as $y = \log(x)$?
3. What is the domain of the function $y = e^x$?
4. How does the domain of $y = a^x$ change if the base $a$ is negative or 0?
5. What is the behavior of the function $y = \left(\frac{1}{5}\right)^x$ as $x \to \infty$?

Tip: When analyzing the domain of exponential functions, remember that they are defined for all real values of $x$ as long as the base is positive and not equal to 1.