Answer:
To find the inverse of:
[tex]f (x)=\dfrac{4}{x-2}-1[/tex]
Set the function to y:
[tex]\implies y=\dfrac{4}{x-2}-1[/tex]
Rearrange to make x the subject:
[tex]\implies y+1=\dfrac{4}{x-2}[/tex]
[tex]\implies (y+1)(x-2)=4[/tex]
[tex]\implies xy-2y+x-2=4[/tex]
[tex]\implies xy+x=2y+6[/tex]
[tex]\implies x(y+1)=2y+6[/tex]
[tex]\implies x=\dfrac{2y+6}{y+1}[/tex]
Swap x and y:
[tex]\implies y=\dfrac{2x+6}{x+1}[/tex]
Change y to the inverse of the function sign:
[tex]\implies f[/tex][tex]\:^{-1}(x)=\dfrac{2x+6}{x+1}[/tex]
Rewrite g(x) as a fraction:
[tex]g(x)=\dfrac{3}{x+2}-2[/tex]
[tex]\implies g(x)=\dfrac{3}{x+2}-\dfrac{2(x+2)}{x+2}[/tex]
[tex]\implies g(x)=\dfrac{3-2(x+2)}{x+2}[/tex]
[tex]\implies g(x)=\dfrac{3-2x-4}{x+2}[/tex]
[tex]\implies g(x)=-\dfrac{2x+1}{x+2}[/tex]
Therefore, as the inverse of f(x) ≠ g(x), the functions are NOT inverses of each other
