Let's examine the provided image about exponential functions and their properties. The function under discussion appears to be $f(x) = a(b^x)$, with the assumption that $a > 0$. We are to identify which statements about this function are true.

Breakdown of the function $f(x) = a(b^x)$:

1. Range of the exponential function: Since $a > 0$ and $b^x$ is always positive for any real $x$ (because an exponential function never crosses zero), the range of $f(x)$ is $(0, \infty)$, i.e., all positive real numbers. Thus, the function will never take on negative values or zero.

2. Domain of the exponential function: The domain of any exponential function is all real numbers because the function $b^x$ is defined for all real $x$, regardless of $b$ as long as $b > 0$.

3. Horizontal Asymptote: Exponential functions generally have a horizontal asymptote. In this case, as $x \to -\infty$, $b^x \to 0$, meaning the function approaches $y = 0$ but never reaches it. Therefore, the horizontal asymptote is the line $y = 0$.

Now, let's evaluate each statement in the list:

1. The range of the exponential function is $f(x) > 0$:

   • This is true since $f(x) = a(b^x)$ for positive $a$ and $b^x > 0$, meaning $f(x) > 0$.

2. The domain of the exponential function is all real numbers:

   • This is true, as the exponential function $b^x$ is defined for all real $x$.

3. The horizontal asymptote is the line $y = 0$:

   • This is true, as explained above. The function approaches zero as $x \to -\infty$, so the asymptote is $y = 0$.

4. The horizontal asymptote is the line $x = 0$:

   • This is false. The horizontal asymptote refers to a line parallel to the x-axis, so it should be $y = 0$, not $x = 0$.

5. The domain of the exponential function is $x > 0$:

   • This is false. The domain is all real numbers, not restricted to $x > 0$.

6. The horizontal asymptote is the point $(0, a)$:

   • This is false. The horizontal asymptote is a line, not a point, and it's $y = 0$, not $(0, a)$.

7. The range of the exponential function is all real numbers:

   • This is false. The range is $(0, \infty)$, not all real numbers.

Correct statements:

• The range of the exponential functions is $f(x) > 0$.
• The domain of the exponential functions is all real numbers.
• The horizontal asymptote is the line $y = 0$.

Would you like more details on any of these concepts or further explanations?

Related questions:

1. How does changing the value of $a$ affect the graph of $f(x) = a(b^x)$?
2. What happens to the graph if $b$ is a value between 0 and 1?
3. How can we find the horizontal asymptote of other types of functions?
4. What is the significance of the domain and range in real-world exponential models?
5. How do transformations such as vertical shifts impact the range and asymptotes of an exponential function?

Tip:

Remember, the horizontal asymptote of an exponential function always corresponds to the value that the function approaches but never quite reaches as $x \to -\infty$.