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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!

Identify the domain, range, and asymptotes for f(x) = 23^x respectively.

Explore how to identify the domain, range, and asymptotes for the exponential function f(x) = 23^x. Learn the properties of exponential functions, including their behavior and horizontal asymptote at y=0. Suitable for high school math students.

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