Answer
a) $\dfrac{amx+mb}{cx+d}$
b) $\dfrac{amx+mb}{cx+d}$
c)
The domain of $(f\circ g) (x)$ is $\mathbb{R}$, $x\ne \frac{-d}{cm}$
The domain of $(g\circ f) (x)$ is $\mathbb{R}$, $x\ne \frac{-d}{c}$
d) $\dfrac{amx+b}{cmx+d}=\dfrac{amx+mb}{cx+d}$
Work Step by Step
We have the composite functions:
a) $(f\circ g) (x) =f[g(x)]=\dfrac{amx+b}{cmx+d}$
b) $(g\circ f) (x) = g[f(x)]=\dfrac{amx+mb}{cx+d}$
c) The domain of both composite functions is all real numbers, except that we cannot divide by zero for certain $x$ values.
Thus, we have:
$cmx+d \ne 0$
$x\ne \frac{-d}{cm}$
and
$cx+d \ne 0$
$x \ne \frac{-d}{c}$
Thus the domain of $(f\circ g) (x)$ is $\mathbb{R}$, $x\ne \frac{-d}{cm}$
And the domain of $(g\circ f) (x)$ is $\mathbb{R}$, $x\ne \frac{-d}{c}$
d) The functions $f(x)$ and $g(x)$ are known as inverse functions. So, we can equate $(f\circ g) (x)=(g\circ f) (x)$ or, $f[g(x)]=g[f(x)]$ as follows:
$\dfrac{amx+b}{cmx+d}=\dfrac{amx+mb}{cx+d}$
