FInd the inverse thanks in advance

Question

Find the inverse of 𝑓(𝑥) = 1 / (𝑥+7) and then verify your result .

(Show that (𝑓 ∘ 𝑓 −1 )(𝑥) = 𝑥 and (𝑓 −1 ∘ 𝑓)(𝑥) = 𝑥.)

Answer 1

We find inverses by solving for x in terms of y.

y = 1/(x+7)

Multiply both sides by x+7

y(x+7) = 1

xy + 7y = 1

xy = 1 - 7y

x = (1 - 7y)/y

x = 1/y - 7

Thus, the inverse is f^-1(x) = 1/x - 7

To check (f º f^-1)(x) = f(1/x - 7) = 1/(1/x - 7 + 7) = 1/(1/x) = x

(f^-1 º f)(x) = f^-1 (1/(x+7)) = 1/(1/(x+7)) - 7 = x + 7 - 7 = x