Final answer:
Both functions f and g are positive in the interval (0, 4). Function f is a logarithmic function with a vertical asymptote at x = 0 and an x-intercept at (4,0), while g(x) = log2(x³) - 2 is positive for x > 2. Therefore, the overlap interval where both are positive is (0, 4).
Step-by-step explanation:
To find the intervals where both functions f and g are positive, we need to analyze the conditions for each function separately and then find their overlap. Given that function f has a vertical asymptote at x = 0 and an x-intercept at (4,0), it is a logarithmic function that is only defined and positive for x > 0. The fact that it is decreasing for x > 0 further confirms it will be positive for x in the range (0, 4), because it crosses the x-axis at 4 and then continues to decrease.
Function g, given by the equation g(x) = log2(x³) - 2, will be positive where log2(x³) > 2. Using the property of logarithms, this inequality can be expressed as x³ > 2² or x³ > 4, which means x > √[3]{4} (the cube root of 4). Since the cube root of 4 is less than 2, g(x) is positive for x > 2. Therefore, the interval where both f(x) and g(x) are positive is (2, 4).
However, we were given the intervals: 1) (0, 4) 2) (4, [infinity]) 3) (0, [infinity]) 4) Cannot be determined. The only interval that fits the condition for both functions to be positive is (0, 4); since after x = 4, function f would not be positive as it has an x-intercept at (4,0).