Final answer:
The solution to the inequality f(x) > g(x) is x ∈ (0, 1) ∪ (2, 3].
Step-by-step explanation:
In order to solve the inequality f(x) > g(x), we need to first evaluate the functions f(x) and g(x). The function f(x) is the floor function, which means it rounds down to the nearest integer. The function g(x) is defined as g(x) = (3/4)x.
In this case, since f(x) rounds down to the nearest integer and g(x) is an increasing function, we can start by finding the integer values of x where the inequality holds true. We will then consider the values between these integers where the inequality also holds true.
Let's solve the inequality step-by-step:
For integer values of x, we have:
f(0) = 0 and g(0) = 0
f(1) = 1 and g(1) = 0.75
f(2) = 2 and g(2) = 1.5
f(3) = 3 and g(3) = 2.25
Between the integers, we can evaluate the inequality for values of x using the definitions of f(x) and g(x). For example, for x = 0.5: f(0.5) = 0 and g(0.5) = 0.375. Since f(x) > g(x) holds true, x = 0.5 is a solution.
Therefore, the solution to the inequality f(x) > g(x) is x ∈ (0, 1) ∪ (2, 3]