Since, [tex]h(x)= (g \circ f)(x)= \frac{1}{(x+3)^2}[/tex],
We have to determine the possible decomposition of h(x).
1. Consider [tex]f(x)=x^2, g(x)=x+3[/tex]
[tex](g\circ f)(x) = g(f(x))=g(x^2)=x^2+3[/tex] which is not equal to the given h(x). This option is not correct.
2. Consider [tex]f(x) = \frac{1}{x} , g(x)=x+3[/tex]
[tex](g\circ f)(x) = g(f(x))=g(\frac{1}{x})=\frac{1}{x}+3 =\frac{1+3x}{x}[/tex] which is not equal to the given h(x). This option is not correct.
3. Consider [tex]f(x)= x+3 , g(x)= \frac{1}{x^2}[/tex]
[tex](g\circ f)(x) = g(f(x))=g((x+3))= \frac{1}{(x+3)^2}[/tex] which is equal to the given h(x). This option is correct.
So, the correct decomposition of h(x) are f(x) = (x+3) and g(x)=[tex]\frac{1}{x^2}[/tex].
Therefore, Option 3 is the correct option.
