Given the functions p(x) and q(x) defined as:
[tex]\begin{gathered} p(x)=x^2+3 \\ q(x)=\sqrt[]{x+2} \end{gathered}[/tex]
We can use the definition of composite functions:
[tex](f\circ g)(x)=f(g(x))[/tex]
Then, to calculate (p o q)(2) = p(q(2)), we need to calculate q(2) first:
[tex]q(2)=\sqrt[]{2+2}=\sqrt[]{4}=2[/tex]
Using this result on the composition:
[tex]\begin{gathered} (p\circ q)(2)=p(q(2))=p(2)=2^2+3=4+3 \\ \Rightarrow(p\circ q)(2)=7 \end{gathered}[/tex]
Now, for (q o p)(2) = q(p(2)), we already calculate p(2) = 7. Then:
[tex]\begin{gathered} (q\circ p)(2)=q(p(2))=q(7)=\sqrt[]{7+2}=\sqrt[]{9} \\ \Rightarrow(q\circ p)(2)=3 \end{gathered}[/tex]
