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 $b > 0$.