Question

If h(x)=(fog)(x) and h(x)=3 sqrt x+3, find g(x) if f(x) =3 sqrt x+2

Answer 1

Answer:

$g(x)=x+\frac{1}{9}+\frac{2\sqrt{x}}{3}$

Step-by-step explanation:

Given functions: h(x) = (fog)(x)   , h(x) = 3√x + 3   and f(x) = 3√x + 2

To find: function g(x)

Consider,

(fog)(x) = h(x)

f( g(x) ) = h(x)

$3\sqrt{g(x)}+2=3\sqrt{x}+3$

$3\sqrt{g(x)}=3\sqrt{x}+1$

$\sqrt{g(x)}=\frac{3\sqrt{x}+1}{3}$

$\sqrt{g(x)}=\sqrt{x}+\frac{1}{3}$

$g(x)=(\sqrt{x}+\frac{1}{3})^2$

$g(x)=(\sqrt{x})^2+(\frac{1}{3})^2+2\times\frac{1}{3}\times\sqrt{x}$

$g(x)=x+\frac{1}{9}+\frac{2\sqrt{x}}{3}$

Therefore, $g(x)=x+\frac{1}{9}+\frac{2\sqrt{x}}{3}$

Answer 2

Answer:

$g(x)=x+1$

Step-by-step explanation:

Consider the functions

$h(x)=3\sqrt{x+3}$

$f(x)=3\sqrt{x+2}$

It is given that

$h(x)=(f\circ g)(x)$

Using the composition of functions, we get

$h(x)=f(g(x))$

$3\sqrt{x+3}=3\sqrt{(g(x)+2}$             $[\because h(x)=3\sqrt{x+3}, f(x)=3\sqrt{x+2}]$

Divide both sides by 3.

$\sqrt{x+3}=\sqrt{(g(x)+2}$

Taking square on both sides.

$x+3=g(x)+2$

Subtract 2 from both sides.

$x+3-2=g(x)$

$x+1=g(x)$

Interchange the sides.

$g(x)=x+1$

Therefore, the required function is g(x)=x+1.