Final answer:
To meet the requirements, a linear function f(x) = x + 1 and an exponential function g(x) = 2^x can be used, where f(x) will be greater than g(x) between 0 and 4, f(0) = 1, and g(4) = 16.
Step-by-step explanation:
To satisfy the given conditions, we need to construct a linear function f(x) and an exponential function g(x) with specific characteristics. The conditions are that f(x) should be greater than g(x) for values of x between 0 and 4, f(0) = 1, and g(4) = 16.
For the linear function f(x), since we want f(0) = 1, this tells us that the y-intercept is 1. Because we want the linear function to be consistently greater than the exponential function between 0 and 4, and knowing that exponential functions grow rapidly, we can select a slope for the linear function that ensures it remains above g(x) in the specified interval. A simple option is f(x) = x + 1. This satisfies f(0) = 1 and also ensures f(x) will be greater than g(x) initially.
For the exponential function g(x), we know that g(4) = 16. Representing this in the form of g(x) = ab^x, and since exponential functions have a constant base, we can find b by solving 16 = ab^4. Assuming the simplest scenario where the initial value a = 1, the base b would be the 4th root of 16, which is 2. Hence, g(x) = 2^x satisfies the condition g(4) = 16.
When plotting these on a graph, we'll see that initially f(x) is above g(x), but eventually, due to the nature of exponential growth, g(x) will surpass f(x). For this particular case, f(x) is greater than g(x) only until roughly x = 3.