function of a function (composite function) with a function not equal to 3

Question

Hello I have the question functions g and h are defined by

g(x)=6/3-x ; x ≠ 3

h(x)=5x+2

I have never come across this type of question where a function has this sign in it ≠

I answered the following question not taking this into account.

A. determine function of g(h(x))

replace x with hx

6/3-(5x+2)

expand bracket -(5x+2)

=-5x-2

=3-5x-2

=-5x-1

g(h(x))=6/-5x-1

B. determine function of h(g(x))

replace x with gx

5(6/3-x)+2

expand 5x6=30

=(30/3-x)+2

h(g(x))=(30/3-x)+2

How do I incorporate the x≠3 into my answer? do I just add it onto the end of g(h(x)) answer?

Answer 1

$x \ne 3$ indicates that the domain of $g$ does not include the value $x=3$.

Indeed, $g(x) = \dfrac {6}{3-x}$ is undefined at $x=3$.

Basically, the rest of the question follows from what you have done, but for $g(h(x))$, the function is not defined for $h(x) = 3$, i.e. $x = -\frac15$, while $h(g(x))$ is not defined for $x=3$.