The answer is:
The statement A "Function f is increasing, but g is decreasing."
<h2>Why?</h2>To solve the problem and select the true statement, we need to compare both functions, using the inputs and outputs of the function "f" and the graph of the function "g" and then, discard each option.
Discarding we have:
A - Function f is increasing but g is decreasing: True.
From the shown table, we can see that the outputs of the function "f" are increasing, starting from 1.25 to 5, so, we can se that the function is increasing.
From the graph, we can see that the function "g" is the opposite of the function "f" since its outputs are decreasing, meaning that the function is decreasing.
Hence, the function "f" is increasing, but g is decreasing.
So, the statement A. "Function f is increasing, but g is decreasing." is true.
B - Both functions have positive x-intercepts.: False.
When a function intercepts the axis "x" we should be able to appreciate at least one output (y) equal to 0. We must remember that when a function intercepts the x-axis the y-value tends to 0.
Hence, the function f does not intercept the x-axis, also, we can see from the graph of the function g that there is not any x-axis intercept.
So, the statement B. "Both functions have positive x-intercepts." is not true.
C - The average rate of change of f is slower than the average rate of change of g over the interval [0, 2].: False.
From the graph of the function g, we can obtain its inputs and outputs and express it as points, and then, calculate its average of change, so, calculating we have:
Using the points: (0,2) and (2,5) we have:
We have that the average rate of change is negative since the function is decreasing, and it's equal to 0.75.
Now, calculating the average of rate of the function f for the same interval, we have:
We have that the average rate of change is positive since the function is decreasing. and it's equal to 1.5
Hence, we have that the average rate of change of f is faster than the average rate of change of g over the interval [0, 2].
So, the statement C. "The average rate of change of f is slower than the average rate of change of g over the interval [0, 2]." is false.
D - The y-intercept of f is greater than the y-intercept of g: False:
When a function intercepts the y-axis, the "x" coordinate of the interception point tends to 0.
For f, we have that the y-intercept is located at "2", we can find the y-intercept using the point (0,2) where "x" is equal to 0, and "y" is equal to 2.
For g, we also have that the y-intercept is located at "2", we can find the y-intercept using the point (0,2) where "x" is equal to 0, and "y" is equal to 2.
Hence, both y-intercepts are equal.
So, the statement D. "the y-intercept of f is greater than the y-intercept of g." is false.
Therefore, we have that correct answer is:
The statement A is true.
Have a nice day!