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Question

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

Answer 1

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)