The function given is $f(x) = 23^x$. Let's break down its properties:

1. Domain: The domain of an exponential function $f(x) = a^x$ (where $a > 0$) is all real numbers, i.e., $(-\infty, \infty)$. So, the domain is $(-\infty, \infty)$.

2. Range: For an exponential function like $f(x) = 23^x$, the function never touches 0, but it approaches 0 as $x \to -\infty$ and grows infinitely large as $x \to \infty$. Therefore, the range is $(0, \infty)$.

3. Asymptotes:

   • Horizontal Asymptote: As $x \to -\infty$, $23^x \to 0$, so there is a horizontal asymptote at $y = 0$.
   • Vertical Asymptote: There is no vertical asymptote for an exponential function like this because the function is defined for all real $x$.

Correct Answer:

Looking at the options:

• The correct domain is $(-\infty, \infty)$,
• The correct range is $(0, \infty)$,
• The horizontal asymptote is $y = 0$.

Thus, the correct answer is Option C.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How would the graph of $f(x) = 23^x$ change if the base were smaller than 1?
2. What would the asymptotes be for a function $f(x) = -23^x$?
3. How does the behavior of $23^x$ compare with that of $e^x$ (the natural exponential function)?
4. What is the significance of horizontal asymptotes in exponential functions?
5. How do logarithmic functions relate to the behavior of exponential functions?

Tip: Exponential functions always have a horizontal asymptote, but vertical asymptotes occur in functions like logarithms!