Question

Find the quotient of the identity function by the reciprocal function.

Answer 1

Let $f:R\to R:f\left(x\right)$ and $g:R-\left\{0\right\}\to R:g\left(x\right)=\frac{1}{x}$ be the identity function and the reciprocal function respectively.

Now, $\text{dom}\left(\frac{f}{g}\right)=\text{dom (f)}\cap \text{dom (g)}-\{x:g(x)=0\}$

and $\{x:g(x)=0\}=\{x:\frac{1}{x}=0\}=\varphi$

$\therefore \text{dom}\left(\frac{f}{g}\right)=[R\cap R-\{0\left\}\right]-\varphi =R-\left[0\right]$

So, $\frac{f}{g}:R-\left\{0\right\}\to R:\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}=\frac{x}{\frac{1}{x}}=x^2$

Hence, $\left(\frac{f}{g}\right)=x^2$ for all $xϵR-\left\{0\right\}$.