Question
3 of 33 of 3 Items
Feature
A logarithmic function is graphed on a coordinate grid. The domain is left-parenthesis 0 comma infinity right-parenthesis. The function decreases over the whole domain passing through left-parenthesis 1 comma 0 right-parenthesis.
Question
Use the graph to answer the question.
Which of the following functions is represented by the graph?
(1 point)
Responses
f left parenthesis x right parenthesis equals log subscript 4 baseline x
Image with alt text: f left parenthesis x right parenthesis equals log subscript 4 baseline x
f left parenthesis x right parenthesis equals log subscript one fourth baseline x
Image with alt text: f left parenthesis x right parenthesis equals log subscript one fourth baseline x
f left parenthesis x right parenthesis equals log subscript 4 baseline left parenthesis negative x right parenthesis
Image with alt text: f left parenthesis x right parenthesis equals log subscript 4 baseline left parenthesis negative x right parenthesis
f left parenthesis x right parenthesis equals log subscript one fourth baseline left parenthesis negative x right parenthesis
Feature
A logarithmic function is graphed on a coordinate grid. The domain is left-parenthesis 0 comma infinity right-parenthesis. The function decreases over the whole domain passing through left-parenthesis 1 comma 0 right-parenthesis.
Question
Use the graph to answer the question.
Which of the following functions is represented by the graph?
(1 point)
Responses
f left parenthesis x right parenthesis equals log subscript 4 baseline x
Image with alt text: f left parenthesis x right parenthesis equals log subscript 4 baseline x
f left parenthesis x right parenthesis equals log subscript one fourth baseline x
Image with alt text: f left parenthesis x right parenthesis equals log subscript one fourth baseline x
f left parenthesis x right parenthesis equals log subscript 4 baseline left parenthesis negative x right parenthesis
Image with alt text: f left parenthesis x right parenthesis equals log subscript 4 baseline left parenthesis negative x right parenthesis
f left parenthesis x right parenthesis equals log subscript one fourth baseline left parenthesis negative x right parenthesis
Answers
GPT-4o mini
To determine which function is represented by the graph described, we need to consider the properties of logarithmic functions:
1. **Domain**: The domain of logarithmic functions is typically limited to positive x-values since the logarithm of non-positive values is undefined. Given that the domain is (0, ∞), we know the function must be a standard logarithmic function defined for positive inputs.
2. **Behavior**: You mentioned that the graph is decreasing throughout its domain. For logarithmic functions, the base of the logarithm determines whether the function is increasing or decreasing:
- If the base is greater than 1 (e.g., log_4(x)), the function is increasing.
- If the base is between 0 and 1 (e.g., log_(1/4)(x)), the function is decreasing.
3. **Point of Interest**: The function passes through the point (1, 0). This means that f(1) = 0, which is consistent with logarithmic properties since log_b(1) = 0 for any positive base b.
Given these points:
- **f(x) = log_4(x)**: This function is increasing because the base (4) is greater than 1.
- **f(x) = log_(1/4)(x)**: This function is decreasing because the base (1/4) is less than 1. It passes through the point (1, 0).
- **f(x) = log_4(-x)** and **f(x) = log_(1/4)(-x)**: These functions are not valid over the domain of (0, ∞) because they involve negative inputs.
Based on the analysis above, the function that is represented by the graph is:
**f(x) = log_(1/4)(x)**
1. **Domain**: The domain of logarithmic functions is typically limited to positive x-values since the logarithm of non-positive values is undefined. Given that the domain is (0, ∞), we know the function must be a standard logarithmic function defined for positive inputs.
2. **Behavior**: You mentioned that the graph is decreasing throughout its domain. For logarithmic functions, the base of the logarithm determines whether the function is increasing or decreasing:
- If the base is greater than 1 (e.g., log_4(x)), the function is increasing.
- If the base is between 0 and 1 (e.g., log_(1/4)(x)), the function is decreasing.
3. **Point of Interest**: The function passes through the point (1, 0). This means that f(1) = 0, which is consistent with logarithmic properties since log_b(1) = 0 for any positive base b.
Given these points:
- **f(x) = log_4(x)**: This function is increasing because the base (4) is greater than 1.
- **f(x) = log_(1/4)(x)**: This function is decreasing because the base (1/4) is less than 1. It passes through the point (1, 0).
- **f(x) = log_4(-x)** and **f(x) = log_(1/4)(-x)**: These functions are not valid over the domain of (0, ∞) because they involve negative inputs.
Based on the analysis above, the function that is represented by the graph is:
**f(x) = log_(1/4)(x)**