Determine whether the following function is one-to-one on its domain. If it is one-to-one, find its inverse function f−1(x). If it is not-one-to-one, write "undefined"

Question

f(x)=x^2−5

f^-1(x)=?

Answer 1

Suppose there are two DIFFERENT values A and B such that

A^2 - 5 = B^2 - 5

Then

A^2 = B^2

A^2 - B^2 = 0

(A+B)(A-B) = 0

A-B = 0 or A+B=0

A=B or A = -B

A=B proves the function is one to one.

For A = -B,

f(-B) = (-B)^2 - 5 = B^2 - 5 = f(B)

which holds because it is an even function.

Therefore the function is one-to-one.

THe inverse is

x = y^2 - 5

x+5 = y^2

y = +or- sqrt( x+5)