Inverse functions

Question

7

Let 

Find a domain on which  is one-to-one and non-decreasing.

Find the inverse of  restricted to this domain.

 

Answer 1

follow the recipe to find the inverse.

1. change f(x) to y and rewrite. y = (x−3)²
2. switch the x and y x = (y−3)²
3. solve for y √x = √(y−3)²

√x = y - 3

√x + 3 = y or y = 3 + √x

now rewrite the y as f^-1(x)

f^-1(x) = 3 + √x

the domain of this inverse is only defined when you can take the square root of x and get a real number, so the domain of f^-1(x) = 3 + √x is when x ≥ 0.

also, since you are adding a growing real number (the square root of x) to 3, as x gets bigger, the inverse is always increasing, so the inverse is non-decreasing over its entire domain.