Question

Pre, Calc, question 4, include answer in bold please, thanks!

Answer 1

Given the functions f and g:

$\begin{gathered} f(x)=x+1 \\ g(x)=\sqrt[]{x+2} \end{gathered}$

to find the composition (g o f)(x), we can write the following:

$(g\circ f)(x)=g(f(x))=g(x+1)$

if we evaluate x+1 on g(x), we get:

$\begin{gathered} g(x+1)=\sqrt[]{(x+1)+2}=\sqrt[]{x+1+2}=\sqrt[]{x+3} \\ \Rightarrow(g\circ f)(x)=\sqrt[]{x+3} \end{gathered}$

now, notice that the composition is not defined on the real numbers that are less than -3 (sice we would get a negative inside the square root), therefore, the domain is:

$\begin{gathered} (g\circ f)(x)=\sqrt[]{x+3} \\ D=\lbrack-3,\infty) \end{gathered}$