To find the value of the composed function (u ∘ w) evaluated in 1, we need to compose the functions.
[tex](u\circ w)=u(w(x))[/tex]This means wherever we have x in the function u, we need to replace it with the expression for w(x) as follows:
[tex]\begin{gathered} (u\circ w)=u(w(x))=w(x)^2+7 \\ \\ (u\circ w)=(\sqrt[]{x+8})^2+7 \end{gathered}[/tex]Solving the squared expression:
[tex]\begin{gathered} (u\circ w)=x+8+7 \\ \\ (u\circ w)=x+15 \end{gathered}[/tex]Now we know the composed function. We just need to evaluate it in x = 1, that is, replace 1 wherever the function has an x:
[tex]\begin{gathered} (u\circ w)(1)=1+15 \\ \\ (u\circ w)(1)=16 \end{gathered}[/tex]Now we have the first answer: (u ∘ w)(1) = 16.
To find the value of (w ∘ u)(1) we follow the same logic:
[tex](w\circ u)=w(u(x))[/tex]In the function w, wherever we have an x, we replace it by the expression for u(x):
[tex]\begin{gathered} (w\circ u)=w(u(x))=\sqrt[]{u(x)+8} \\ \\ (w\circ u)=\sqrt[]{(x^2+7)+8}=\sqrt[]{x^2+7+8} \\ \\ (w\circ u)=\sqrt[]{x^2+15} \end{gathered}[/tex]Now we know the composed function (w ∘ u). We just need to evaluate it in x = 1, that is, replace 1 wherever the function has an x:
[tex]\begin{gathered} (w\circ u)(1)=\sqrt[]{1^2+15}=\sqrt[]{1^{}+15} \\ \\ (w\circ u)(1)=\sqrt[]{16} \\ \\ (w\circ u)(1)=4 \end{gathered}[/tex]Now we have the second answer: (w ∘ u)(1) = 4.