finding inverse and verify

Question

Find the inverse of f(x) and then verify your result using (𝑓 ° 𝑓^−1)(𝑥) = 𝑥 and (𝑓^−1 ° 𝑓)(𝑥) = 𝑥.

F(x)=1/x-3

Answer 1

(1) f(x)=1/x-3

let y = f(x)

solve (1) for x = g(y) = f^-1(y)

y+3 = 1/x

x = 1/(y+3)

f^-1(y) = 1/(y+3)

claim f^-1(f(x)) = x

and f(f^-1(x)) = x

proof:

f^-1(f(x)) = f^-1(y) = 1/(f(x)+3) = 1/(1/x-3 + 3) = 1/(1/x) = x

f(f^-1(x)) = f(1/(x+3)) = 1/(1/(x+3)) - 3 = (x+3) - 3 = x

Q.E.D.