Answer: The pair of functions f(x) = 2*¹+1 and g(x) = log₂ (x - 1) - 1 are inverse functions.
Explanation:
To determine if two functions are inverses of each other, we need to check if the composition of the functions gives us the identity function, which is denoted as f(g(x)) = x and g(f(x)) = x.
Let's start by checking f(g(x)):
f(g(x)) = f(log₂ (x - 1) - 1)
To simplify, we substitute g(x) into f(x):
f(g(x)) = f(log₂ (x - 1) - 1) = 2(log₂ (x - 1) - 1) + 1
Next, let's check g(f(x)):
g(f(x)) = g(2*¹+1)
To simplify, we substitute f(x) into g(x):
g(f(x)) = g(2*¹+1) = log₂ (2*¹+1 - 1) - 1
By evaluating both compositions, we can see that f(g(x)) = g(f(x)) = x. Therefore, f(x) = 2*¹+1 and g(x) = log₂ (x - 1) - 1 are inverse functions.