Question

Show f and g are inverse functions. Show analytically and graphically. I don’t know if I’m right but would like to know.

Answer 1

Hello!

To prove that f(x) = 2x - 1 and g(x) = x/2 + 1/2, we can use a composite function. Composite functions are basically (f ∘ g)(x). It combines two functions into one. If they are true inverses, then the answer must be equal to x.

(f ∘ g)(x) = 2(x/2 + 1/2) - 1

(f ∘ g)(x) = x + 1 - 1

(f ∘ g)(x) = x

(g ∘ f)(x) = (2x - 1)/2 + 1/2

(g ∘ f)(x) = x - 1/2 + 1/2

(g ∘ f)(x) = x

Since (g ∘ f)(x) and (f ∘ g)(x) are both equal to x, then the functions of f(x) and g(x) are inverses of each other.

Also, in order for two functions to be inverses, these two functions need to be reflected over the line y = x. In the graph shown below, y = x is in red, y = 2x - 1 is blue, and y = x/2 + 1/2 is green. Looking the graph, you can see they are reflected over the line y = x.

Therefore, the function f(x) = 2x - 1 and g(x) = x/2 + 1/2 are true inverses of each other.