Question

Find $gof$ and $fog$, if $\left(i\right)$ $f\left(x\right)=|x|$ and $g\left(x\right)=|5x-2|$

ii)$f\left(x\right)=8x^3$ and $g\left(x\right)=x^{\frac{1}{3}}$

Answer 1

$\left(i\right)$ $f\left(x\right)=|x|$ and $g\left(x\right)=|5x-2|$

$\therefore \left(gof\right)\left(x\right)=g\left(f\right(x\left)\right)=g\left(|x|\right)=|5|x|-2|$

And $\left(fog\right)\left(x\right)=f\left(g\right(x\left)\right)=f(|5x-2|)=||5x-2||=|5x-2|$
$\left(ii\right)f\left(x\right)=8x^3$ and $g\left(x\right)=x^{\frac{1}{3}}$

$\therefore \left(gof\right)\left(x\right)=g\left(f\right(x\left)\right)=g\left(8x^3\right)=(8x^3{)}^{\frac{1}{3}}=2x$

And $\left(fog\right)\left(x\right)=f\left(g\right(x\left)\right)=f\left(x^{\frac{1}{3}}\right)=8{\left(x^{\frac{1}{3}}\right)}^3=8x$