Sereinserenormg

Answered

2022-01-25

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

Answer & Explanation

Jasmine Herman

Expert

2022-01-26Added 11 answers

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

Now

It may be easier to think about if we gave

So we see that

So

Another answer is to letf

So let

To get

So let

Kingston Gates

Expert

2022-01-27Added 8 answers

Define these functions:

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

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

Then:

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

Then:

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