We are given:
$\eqalign{
& f(x) = x – 1 \cr
& g(x) = {x^2} \cr} $
Given these functions the answer I get is:
$fg(x)=x^2-1$
As the range of $g(x)=x^2$ is always positive this means the domain of the new function is $x \geqslant 0 $
The range of this function is $fg(x) \geqslant – 1$
However this is wrong, the answer in the textbook states that the domain is the set of real numbers, and I know this makes sense as it is a quadratic equation and can accept all values but doesn't the function $g(x)$ who's outputs form the inputs of the compound function $gf(x)$ limit the domain to being $x \geqslant 0 $ ?
Thanks you for your help.
Best Answer
It’s true that the outputs of $g$ are non-negative, but since $f$ can accept any real number as input, it really doesn’t matter what the outputs of $g$ are. The domain of the composite function $f\circ g$ is the set of all acceptable inputs to $f\circ g$; $g$ can accept any real number as input, and the output of $g$ is always acceptable input to $f$, so $f\circ g$ can accept any real number as input. In other words, its domain really is $\Bbb R$.