Sereinserenormg

2022-01-25

How do you write $y=3\sqrt{1+{x}^{2}}$ as a composition of two simpler functions?

Jasmine Herman

Expert

Let $g\left(x\right)$ be the first thing we do if we knew $x$ and started to calculate:
$g\left(x\right)={x}^{2}$
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 letf $\left(x\right)$ be the last thing we would do in calculating $y$.
So let $f\left(x\right)=3x$
To get $y=f\left(g\left(x\right)\right)$ we need $3g\left(x\right)=y$
So let $g\left(x\right)=\sqrt{1+{x}^{2}}$

Kingston Gates

Expert

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

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