# How do you write l(x) = (tan(x^2))^.5 as a composition of two or more functions?

Dec 27, 2015

$l \left(x\right) = f \left(g \left(x\right)\right) = {\left(\tan \left({x}^{2}\right)\right)}^{\frac{1}{2}}$

$g \left(x\right) = \tan \left({x}^{2}\right)$

$f \left(x\right) = {x}^{\frac{1}{2}} = {x}^{.5}$

#### Explanation:

There are multiple way to answer this question but the most simple (in my opinion) is this

We know $l \left(x\right) = {\left(\tan \left({x}^{2}\right)\right)}^{\frac{1}{2}}$

We also know that $l \left(x\right) = f \left(g \left(x\right)\right)$ (definition of composition of function)

We can let the inside function be
$g \left(x\right) = \tan {x}^{2}$

And the outside function $f \left(x\right) = {x}^{\frac{1}{2}} = \sqrt{x}$

$l \left(x\right) = f \left(g \left(x\right)\right) = f \left(\tan {x}^{2}\right) = {\left(\tan \left({x}^{2}\right)\right)}^{\frac{1}{2}} = {\left(\tan \left({x}^{2}\right)\right)}^{.5}$