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

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.
$$\displaystyle{q}\cdot{r}{\left({1}\right)}=?$$
$$\displaystyle{\left({r}\cdot{q}\right)}{\left({1}\right)}=?$$

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