Inverse Function Question

Question

A function is defined on a suitable domain as

f(x) =(x+3)^1/2 -4

show clearly that the inverse function, which exists, can be written in the form

f^-1(x)= (x+p)^2 + q, and state the values of q.

Answer 1

Hello Jack,

Starting with f(x) = (x+3)^1/2 -4

Replace f(x) by y: y = (x+3)^1/2 -4

Swap y and x: x = (y+3)^1/2 -4

Solve for y by adding 4 on both sides: x + 4 = (y+3)^1/2

Square both sides: (x+4)² = (y+3)

Subtract 3 from both sides: (x+4)² -3 = y

Replace y by f^-1(x): f^-1(x) = (x+4)² -3

Matching the answer to f^-1(x)= (x+p)² + q, you can see that p = 4 and q = -3

Cheers.