Answer:
[tex]f^{-1}(x)=x+5.[/tex]
The domain of the function [tex]f^{-1}(x)[/tex] is [tex]x\ge 0[/tex] (greater than or equal to 0).
Step-by-step explanation:
If [tex]x\ge 5,[/tex] then the expression [tex]x-5[/tex] takes values that are greater or equal than 0. Thus,
[tex]f(x)=|x-5|=x-5\text{ for }x\ge 5.[/tex]
To find the inverse function you have to express x in terms of y and then change x into y and y into x:
[tex]y=x-5,\\ \\x=y+5[/tex]
and
[tex]f^{-1}(x)=x+5.[/tex]
The domain of the function [tex]f(x)[/tex] is [tex]x\ge 5[/tex] and the range of the function [tex]f(x)[/tex] is [tex]y\ge 0.[/tex] The domain of the inverse function [tex]f^{-1}(x)[/tex] is the range of the function [tex]f(x),[/tex] hence the domain of the function [tex]f^{-1}(x)[/tex] is [tex]x\ge 0.[/tex]
