Define the functions $f, g$ and $h$ from R to R such that $f(x)=x^2-1, g(x)=\sqrt{x^2+1}$ and…
Define the functions $f, g$ and $h$ from R to R such that $f(x)=x^2-1, g(x)=\sqrt{x^2+1}$ and $h(x)=\left\{\begin{array}{l}0, \text { if } x \leq 0 \\ x, \text { if } x \geq 0\end{array}\right.$
consider the following statements
fog is invertible
$h$ is an identify function
fog is not invertible
$(h \circ f \circ g) x=x^2$
Solution
$f \circ g(x)=f\left(\sqrt{x^2+1}\right)=\left(x^2+1\right)-1=x^2$
$\because$ Codomain of $\operatorname{fog}(x)=\mathrm{R}$
and Range of fog $(x)=[0, \infty)$
$\therefore$ fog is not onto function $\quad \Rightarrow$ fog is not invertible.
Now $(h \circ f \circ g) x=h \circ[f \circ g(x)]=h\left(x^2\right)$
$\because x^2 \geq 0 \forall x \in \mathrm{R}$. Hence, $(h \circ$ fog $)(x)=x^2$
also given, $h(x)=\left\{\begin{array}{ll}0, & \text { if } x \leq 0 \\ x, & \text { if } x \geq 0\end{array}\right.$ is not identity
function because for $x \lt 0, h(x) \neq x$