Suppose that the dunctions q and r are defined as follows. q(x)=x^2+7 r(x)=sqrt(x+8) Find the following. [email protected](1)=? ([email protected])(1)=?

Suppose that the dunctions q and r are defined as follows. q(x)=x^2+7 r(x)=sqrt(x+8) Find the following. [email protected](1)=? ([email protected])(1)=?

Question
Composite functions
asked 2021-01-05
Suppose that the dunctions q and r are defined as follows.
\(\displaystyle{q}{\left({x}\right)}={x}^{{2}}+{7}\)
r(x)=sqrt(x+8)ZSK
Find the following.
\(\displaystyle{q}\circ{r}{\left({1}\right)}=?\)
\(\displaystyle{\left({r}\circ{q}\right)}{\left({1}\right)}=?\)

Answers (1)

2021-01-06
Find the function definition for the first composite function, as follows:
\(\displaystyle{\left({q}\circ{r}\right)}{\left({x}\right)}={q}{\left({r}{\left({x}\right)}\right)}\)
\(\displaystyle={q}{\left(\sqrt{{{x}+{8}}}\right)}\)
\(\displaystyle={\left(\sqrt{{{x}+{8}}}^{{2}}+{7}\right.}\)
\(\displaystyle={x}+{8}+{7}\)
\(\displaystyle={x}+{15}\)
Find the function definition for the second composite function, as follows:
\(\displaystyle{\left({r}\circ{q}\right)}{\left({x}\right)}={r}{\left({q}{\left({x}\right)}\right)}\)
\(\displaystyle={r}{\left({x}^{{2}}+{7}\right)}\)
\(\displaystyle=\sqrt{{{\left({x}^{{2}}+{7}\right)}+{8}}}\)
\(\displaystyle=\sqrt{{{x}+{15}}}\)
Find the value of the first composite function at x=1, as follows:
\(\displaystyle{\left({q}\circ{r}\right)}{\left({1}\right)}={\left({1}\right)}+{15}={16}\)
Find the value of the second composite function at x=1, as follows:
\(\displaystyle{\left({r}\circ{q}\right)}{\left({1}\right)}=\sqrt{{{1}^{{2}}+{15}}}\)
\(\displaystyle=\sqrt{{16}}\)
=4
0

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