Question

Suppose that the functions p and q are defined as follows in the attachment

Answer 1

$\begin{gathered} (q\circ p)(-5)=50 \\ (p\circ q)(-5)=-24 \end{gathered}$

1) In this question, we need to write out two composite functions.

2) So let's start with the following composite functions and then evaluate them for x=-5:

$\begin{gathered} (q\circ p)(x)=\mleft(x-2\mright)^2+1 \\ (p\circ q)(x)=-(x^2+1)+2=-x^2-1+2=-x^2+1 \end{gathered}$

2.2) Now, we can evaluate those two composite functions for x=-5

$\begin{gathered} (q\circ p)(x)=\mleft(x-2\mright)^2+1 \\ (q\circ p)(-5)=\mleft(\mleft(-5\mright)-2\mright)^2+1=50 \\ \\ (p\circ q)(x)=-(x^2+1)+2 \\ (p\circ q)(-5)=-((-5)^2+1)+2=-(26)+2=-24 \end{gathered}$

3) Hence, the answers are:

$\begin{gathered} (q\circ p)(-5)=50 \\ (p\circ q)(-5)=-24 \end{gathered}$