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Let's analyze each statement to determine which are true for the exponential function \( f(x) = a(b)^x \) with \( a > 0 \) and \( b > 0 \).

1. **The Domain of the exponential functions is \( x > 0 \)**: This is incorrect. The domain of exponential functions is all real numbers (\( x \in \mathbb{R} \)).

2. **The Range of the exponential functions is \( f(x) > 0 \)**: This is correct. Since \( a > 0 \) and \( b > 0 \), the output of the function will always be positive, meaning \( f(x) > 0 \).

3. **The Horizontal Asymptote is the line \( y = 0 \)**: This is correct. For exponential functions, as \( x \) approaches negative infinity, \( f(x) \) approaches zero but never reaches it, making \( y = 0 \) the horizontal asymptote.

4. **The Domain of the exponential functions is All Real Numbers**: This is correct. Exponential functions have a domain of \( x \in \mathbb{R} \).

5. **The Horizontal Asymptote is the point \( (0, a) \)**: This is incorrect. A horizontal asymptote is a line, not a point, and in this case, it’s \( y = 0 \), not \( (0, a) \).

6. **The Horizontal Asymptote is the line \( x = 0 \)**: This is incorrect. The horizontal asymptote is \( y = 0 \), not \( x = 0 \).

7. **The Range of the exponential functions is All Real Numbers**: This is incorrect. The range is \( f(x) > 0 \), not all real numbers.

Based on this analysis, the correct answers are:
- **The Range of the exponential functions is \( f(x) > 0 \)**.
- **The Horizontal Asymptote is the line \( y = 0 \)**.
- **The Domain of the exponential functions is All Real Numbers**.

Do you want more details on any of these points, or have any questions?

### Related Questions:
1. What would happen to the function's properties if \( a \) or \( b \) were negative?
2. How do the domain and range of exponential functions compare to those of logarithmic functions?
3. Why is \( y = 0 \) considered a horizontal asymptote for most exponential functions?
4. Can an exponential function have a horizontal asymptote other than \( y = 0 \)?
5. How does changing the value of \( b \) (growth vs. decay) affect the graph of the function?

### Tip:
Remember that exponential functions of the form \( f(x) = a(b)^x \) always have a domain of all real numbers and usually a horizontal asymptote at \( y = 0 \) when \( b > 0 \).

According to the Mini-Lesson, which of the following are TRUE regarding \( f(x) = a(b)^x \)? Check all that apply. Assume \( a > 0 \).

Explore which properties are true for the exponential function \( f(x) = a(b)^x \) with \( a > 0 \) and \( b > 0 \). This lesson covers the domain, range, and asymptote properties of exponential functions, suitable for high school students in Grades 9-12.

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