f is defined by $f:x\mapsto x-2$ and g is a function such that $g\circ f:x\mapsto\frac{1}{11-3x}$ , $x≠ \frac{11}{3}$ Find the function g

Question

The function $f$ is defined by $f:x\mapsto x-2$ and another function $g$ is such that

$$g\circ f:x\mapsto \frac{1}{11-3x} , x≠\frac{11}{3}$$

Find the function $g$.

Answer 1

If you mean $gf$ as a composition, then $g(x)=\frac{1}{5-3x}$.

Answer 2

$$ f(x) = x-2 $$ $$ g(f(x)) = \frac{1}{11-3x} $$ $$ g(x-2) = \frac{1}{11-3x} $$ Substitute $t = x-2$ $$ g(t) = \frac{1}{11-3(t+2)} = \frac{1}{5-3t} $$