Question

Find the product of the identity function by the modulus function.

Answer 1

Let $f:R\to R:f\left(x\right)=x$ and $g:R\to R:g\left(x\right)=|x|$ be the identity function and the modulus function respectively.

Then, $\text{dom (fg)}=\text{dom (f)}\cap \text{dom (g)}=R\cap R=R$

$\therefore \left(fg\right):R\to R:\left(fg\right)\left(x\right)=f\left(x\right).g\left(x\right)$

Now, $\left(fg\right)\left(x\right)=f\left(x\right).g\left(x\right)=x.|x|$

$=x.\{\begin{array}{c}x,\text{when}x\ge 0\\ -x,\text{when}x<0\end{array}=\{\begin{array}{c}{x}^2,\text{when}x\ge 0\\ -x^2,\text{when}x<0\end{array}$

Hence, $\left(fg\right)\left(x\right)=\{\begin{array}{c}{x}^2,\text{when}x\ge 0\\ -x^2,\text{when}x<0\end{array}$