Question

Suppose f(x)= 5-x/3 (fraction) and g(x) = 5-3x. Use composition of functions to show f(x) and g(x) are inverses of each other.Write a conclusion statement to complete your proof. solve step by step show work

a) f(g(x)) = x and g(f(x)) = x
b) f(x) = g(x) and g(x) = f(x)
c) f(g(x)) = g(f(x)) = x
d) f(x) + g(x) = x

Answer 1

To prove that f(x) and g(x) are inverses of each other, we need to show that when we compose them, we get x as the output. By substituting the expressions for f(x) and g(x) and simplifying, it can be shown that f(g(x)) = g(f(x)) = x.

Step-by-step explanation:

To show that f(x) and g(x) are inverses of each other, we need to show that when we compose them, we get x as the output.

a) f(g(x)) = x: Substitute g(x) into f(x): f(g(x)) = f(5-3x) = 5 - (5-3x)/3 = 5 - (5/3) + (x/3) = x, which proves that f(g(x)) = x.

b) g(f(x)) = x: Substitute f(x) into g(x): g(f(x)) = g(5-x/3) = 5 - 3(5-x/3) = 5 - 3(5) + x = x, which proves that g(f(x)) = x.

Therefore, the correct answer is c) f(g(x)) = g(f(x)) = x.