Question

Find f(x) and g(x) so the function can be expressed as y = f(g(x)).

y = Two divided by x squared. + 3

Answer 1

Answer: g(x) =
`x^2`

and f(x) =
`(2)/(x+3)`

Explanation:

`y=(2)/(x^2+3)`

To find f(x) and g(x) such that

`y=f(g(x))=(2)/(x^2+3)`,

Then only one possible answer is g(x) =
`x^2`

and f(x) =
`(2)/(x+3)` such that

`\text{the composite function will be }\ f(g(x))=f(x^2)=(2)/((x^2)+3)=(2)/(x^2+3)`

Thus, g(x) =
`x^2`

and f(x) =
`(2)/(x+3)`

Answer 2

y=2/x² +3
let's find f and g
y=f(g(x))=2/x² +3, y-3= 2/x² , so f^-1(y)= y-3= 2/x² = g(x)
so g(x) =2/x²
but y = 2/x² +3, so y= g(x)+3= f(g(x)), consequently, f(x) = x+3

finally f(x) = x+3, and g(x)=2/x²