Question

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

Composite functions
ANSWERED
asked 2021-01-05

Suppose that the dunctions q and r are defined as follows.
\(\displaystyle{q}{\left({x}\right)}={x}^{{2}}+{7}\)
\(r(x)=\sqrt{x+8}\)
Find the following.
\(\displaystyle{q}\cdot{r}{\left({1}\right)}=?\)
\(\displaystyle{\left({r}\cdot{q}\right)}{\left({1}\right)}=?\)

Answers (1)

2021-01-06

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

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