Question

I need to find the composite function with these two equations. I also need to find the domain.

Answer 1

Recall that:

$(f\circ f)(x)=f(f(x)).$

Therefore:

$(f\circ f)(x)=f(\sqrt[]{x+2})=\sqrt[]{\sqrt[]{x+2}+2}.$

Now, the above function is well defined as long as x+2 remains positive, therefore, it is well defined as long as x is greater or equal to -2.

Answer: The domain is:

$\lbrack-2,\infty).$

The composition is:

$(f\circ f)(x)=f(\sqrt[]{x+2})=\sqrt[]{\sqrt[]{x+2}+2}.$