Answer:
Question 1
The function [tex]f(x) = x + 6[/tex] is one-to-one, so it does have an inverse.
The inverse of +6 is -6, so [tex]f^{-1}(x)=x-6[/tex]
Therefore, g(x) is the inverse of f(x).
Question 2
The function [tex]f(x) = -3x -9[/tex] is one-to-one, so it does have an inverse.
To find the inverse, replace f(x) with y:
[tex]\implies y = -3x -9[/tex]
Rearrange the equation to make x the subject:
[tex]\implies y+9= -3x[/tex]
[tex]\implies x=-\dfrac13(y+9)[/tex]
[tex]\implies x=-\dfrac13y-3[/tex]
Replace x with [tex]f^{-1}(x)[/tex] and y with x:
[tex]\implies f^{-1}(x)=-\dfrac13x-3[/tex]
Therefore, g(x) is the inverse of f(x).
