# How do you write  y = 3sqrt(1 + x^2) as a composition of two simpler functions?

Aug 11, 2015

Define these functions:
$g \left(x\right) = 1 + {x}^{2}$
$f \left(x\right) = 3 \sqrt{x}$

Then:
$y \left(x\right) = f \left(g \left(x\right)\right)$

Aug 13, 2015

There is more than one way to do this.

#### Explanation:

Let $g \left(x\right)$ be the first thing we do if we knew $x$ and started to calculate:

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

Now $f$ will be the rest of the calculation we would do (after we found ${x}^{2}$)

It may be easier to think about if we gave $g \left(x\right)$ a temporary name, say $g \left(x\right) = u$

So we see that $y = 3 \sqrt{1 + u}$

So $f \left(u\right) = 3 \sqrt{1 + u}$ and that tells us we want:

$f \left(x\right) = 3 \sqrt{1 + x}$

Another answer is to let $f \left(x\right)$ be the last thing we would do in calculating $y$.

So let $f \left(x\right) = 3 x$

To get $y = f \left(g \left(x\right)\right)$ we need $3 g \left(x\right) = y$

So let $g \left(x\right) = \sqrt{1 + {x}^{2}}$