Final answer:
The function g is an exponential function passing through the points (0,7) and (3,0). By solving for the constants in the general form of an exponential function, we find that the function g(x) is 0. Therefore, the correct statement comparing the two functions on the interval (0, 3) is d) One function is positive on the interval, while the other is negative. Hence, the correct statement comparing the two functions on the interval (0, 3) is d) One function is positive on the interval, while the other is negative.
Step-by-step explanation:
The given information states that the function g is an exponential function passing through the points (0,7) and (3,0). To determine the behavior of the function on the interval (0, 3), we need to understand the properties of exponential functions.
An exponential function has the general form f(x) = a * b^(x-c), where a, b, and c are constants. In this case, the function g passes through the points (0,7) and (3,0). By substituting these points into the general form, we can find the specific equation for g. Let's start with (0,7):
7 = a * b^(0-c) = a * b^(-c) = a / b^c
Next, let's use (3,0):
0 = a * b^(3-c) = a * b^(3-0) = a * b^3
Using these two equations, we can solve for a and b. Dividing the second equation by the first equation, we get:
b^3 / b^(-c) = 0 / 7
b^(3-(-c)) = 0
b^3 * b^c = 0
b^(3+c) = 0
Since any positive number raised to any power will never equal 0, we can conclude that b = 0. Therefore, the function g(x) is g(x) = a * 0^(x-c) = a * 0 = 0. Hence, the correct statement comparing the two functions on the interval (0, 3) is d) One function is positive on the interval, while the other is negative.