The problem states that the function $f(x) = 5^x$ is given, and it asks for the domain of the function.

Solution:

The domain of a function refers to all possible values of $x$ for which the function is defined.

For the exponential function $f(x) = 5^x$:

• The base $5$ is positive and greater than 0.
• The exponent $x$ can take any real number value (positive, negative, or zero), as there are no restrictions on $x$ for an exponential function.

Thus, the domain of $f(x) = 5^x$ is all real numbers.

Correct Answer:

(D) all real numbers

Let me know if you need a deeper explanation or have more questions! Here are some related questions to explore further:

1. What is the range of $f(x) = 5^x$?
2. How does the graph of $5^x$ behave as $x \to \infty$ or $x \to -\infty$?
3. What is the domain and range of $f(x) = 5^{-x}$?
4. How do exponential functions differ from polynomial functions in behavior?
5. What is the derivative of $f(x) = 5^x$, and how is it computed?

Tip:

For any exponential function $f(x) = a^x$, where $a > 0$ and $a \neq 1$, the domain is always all real numbers, and the range is $(0, \infty)$.