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

`g(x)=x+(1)/(9)+(2√(x))/(3)`

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√(g(x))+2=3√(x)+3`

`3√(g(x))=3√(x)+1`

`√(g(x))=(3√(x)+1)/(3)`

`√(g(x))=√(x)+(1)/(3)`

`g(x)=(√(x)+(1)/(3))^2`

`g(x)=(√(x))^2+((1)/(3))^2+2*(1)/(3)*√(x)`

`g(x)=x+(1)/(9)+(2√(x))/(3)`

Therefore,
`g(x)=x+(1)/(9)+(2√(x))/(3)`

Answer 2

`g(x)=x+1`

Explanation:

Consider the functions

`h(x)=3√(x+3)`

`f(x)=3√(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√(x+3)=3√((g(x)+2)`
`[\because h(x)=3√(x+3), f(x)=3√(x+2)]`

Divide both sides by 3.

`√(x+3)=√((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.