Question

Which of the following functions are not identical?

1. $f\left(x\right)=x/x,g\left(x\right)=1$

2. $f\left(x\right)=1,g\left(x\right)={\sin }^{2}x+{\cos }^{2}x$

3. $f\left(x\right)=x,g\left(x\right)={\left(\sqrt{x}\right)}^2$

4. $f\left(x\right)=\log (x-2)+\log (x-3),g\left(x\right)=\log (x-2)(x-3)$

Answer 1

A) $f\left(x\right)=\frac{x}{x},g\left(x\right)=1$

Domain of $f\left(x\right)=R-\left\{0\right\}$

Domain of $g\left(x\right)=R$

$\therefore f\left(x\right)\&g\left(x\right)$ aren't identical

(B) $f\left(x\right)=1,g\left(x\right)=si{n}^{2}x+co{s}^{2}x$

${}^{\prime }{f}^{\prime }{\&}^{\prime }{g}^{\prime }$ are identical

(C) $f\left(x\right)=x,g\left(x\right)=(\sqrt{x}{)}^2$

Domain of $f\left(x\right)$ is $x\in R$

Domain of $g\left(x\right)$ is $x>0$

${\therefore }^{\prime }{f}^{\prime }{\&}^{\prime }{g}^{\prime }$ aren't identical.

(D) $f\left(x\right)=\log (x-2)+\log (x-3)$

Domain of $f\left(x\right)$ is $x\in (3,\infty )$

$g\left(x\right)=\log (x-2)(x-3)$

Domain of $g\left(x\right)$ is $x\in (-\infty ,2)\cup (3,\infty )$

${\therefore }^{\prime }{f}^{\prime }{\&}^{\prime }{g}^{\prime }$ aren't identical.