https://file.aiquickdraw.com/mathbot/file/17295689666216i5sznes.jpg

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 \( x \to -\infty \).

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

Explore which statements about the exponential function f(x) = a(b^x) are true, including discussions on the domain, range, and horizontal asymptotes. This problem is ideal for high school students and covers essential properties of exponential functions with a > 0.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation