Question

Determine the inverse of the function f(x)=|x-5| for x (greater than or equal to) 5. Then complete the inequality for the domain restrictions of the inverse function

F^-1(x)=______
The domain restriction of f^-1(x) is x (greater than or equal to)

Answer 1

`f^(-1)(x)=x+5.`

The domain of the function
`f^(-1)(x)` is
`x\ge 0` (greater than or equal to 0).

Explanation:

If
`x\ge 5,` then the expression
`x-5` takes values that are greater or equal than 0. Thus,

`f(x)=|x-5|=x-5\text{ for }x\ge 5.`

To find the inverse function you have to express x in terms of y and then change x into y and y into x:

`y=x-5,\\ \\x=y+5`

and

`f^(-1)(x)=x+5.`

The domain of the function
`f(x)` is
`x\ge 5` and the range of the function
`f(x)` is
`y\ge 0.` The domain of the inverse function
`f^(-1)(x)` is the range of the function
`f(x),` hence the domain of the function
`f^(-1)(x)` is
`x\ge 0.`