# Given f(x) = x^2, g(x) = x + 7 Find (f @ g)(x). Find the domain of (f @ g)(x). Find (g @ f)(x). Find the domain of (g @ f)(x). Find (f @ f)(x). Find the domain of (f @ f)(x). Find (g @ g)(x). Find the domain of (g @ g)(x).

Question
Composite functions
Given
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}},{g{{\left({x}\right)}}}={x}+{7}$$
Find $$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}.$$
Find the domain of $$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}.$$
Find $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}.$$
Find the domain of $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}.$$
Find $$\displaystyle{\left({f}\circ{f}\right)}{\left({x}\right)}.$$
Find the domain of $$\displaystyle{\left({f}\circ{f}\right)}{\left({x}\right)}.$$
Find $$\displaystyle{\left({g}\circ{g}\right)}{\left({x}\right)}.$$
Find the domain of $$\displaystyle{\left({g}\circ{g}\right)}{\left({x}\right)}.$$

2020-10-22
$$\displaystyle{f}\circ{g{{\left({x}\right)}}}={f{{\left({g{{\left({x}\right)}}}\right)}}}={\left({g{{\left({x}\right)}}}\right)}^{{2}}={\left({x}+{7}\right)}^{{2}}$$
$$\displaystyle={x}^{{2}}+{14}{x}+{49}$$
It is a polunomial so domain is all real numbers
Domain of $$\displaystyle{f}\circ{g{{\left({x}\right)}}}$$ is $$\displaystyle{\left(-\infty,\infty\right)}$$
$$\displaystyle{g}\circ{f{{\left({x}\right)}}}={f{{\left({x}\right)}}}+{7}={x}^{{2}}+{7}$$
It is a polunomial so domain is all real numbers
Domain of $$\displaystyle{g}\circ{f{{\left({x}\right)}}}$$ is $$\displaystyle{\left(-\infty,\infty\right)}$$
$$\displaystyle{f}\circ{f{{\left({x}\right)}}}={\left({f{{\left({x}\right)}}}\right)}^{{2}}={\left({x}^{{2}}\right)}^{{2}}={x}^{{4}}$$
It is a polunomial so domain is all real numbers
Domain of $$\displaystyle{f}\circ{f{{\left({x}\right)}}}$$ is $$\displaystyle{\left(-\infty,\infty\right)}$$
$$\displaystyle{g}\circ{g{{\left({x}\right)}}}={g{{\left({x}\right)}}}+{7}={x}+{7}+{7}={x}+{14}$$
It is a polunomial so domain is all real numbers
Domain of $$\displaystyle{g}\circ{g{{\left({x}\right)}}}$$ is $$\displaystyle{\left(-\infty,\infty\right)}$$

### Relevant Questions

Find the composite functions $$\displaystyle{f}\circ{g}$$ and $$\displaystyle{g}\circ{f}$$. Find the domain of each composite function. Are the two composite functions equal?
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}−{1}$$
g(x) = −x
Find the composite functions $$\displaystyle{f}\circ{g}$$ and $$\displaystyle{g}\circ{f}$$. Find the domain of each composite function. Are the two composite functions equal
f(x) = 3x + 1
g(x) = −x
Given
$$\displaystyle{f{{\left({x}\right)}}}={2}-{x}{2},{g{{\left({x}\right)}}}=\sqrt{{{x}+{2}}}$$
$$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}},{g{{\left({x}\right)}}}=\sqrt{{{1}-{x}}}$$
(a) write formulas for $$\displaystyle{f}\circ{g}$$ and $$\displaystyle{g}\circ{f}$$ and find the
(b) domain and
(c) range of each.
The regular price of a computer is x dollars. Let f(x) = x - 400 and g(x) = 0.75x. Solve,
a. Describe what the functions f and g model in terms of the price of the computer.
b. Find $$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}$$ and describe what this models in terms of the price of the computer.
c. Repeat part (b) for $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}$$.
d. Which composite function models the greater discount on the computer, $$\displaystyle{f}\circ{g}$$ or $$\displaystyle{g}\circ{f}$$?
Find the domain of the coposite function [email protected]

$$\displaystyle{f{{\left({x}\right)}}}=\frac{{5}}{{{x}+{9}}}$$
g(x)=x+6
Compose the following function and state the domain of the composed function.
$$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{x}}-{1}$$
$$\displaystyle{g{{\left({x}\right)}}}=\sqrt{{{1}-{x}^{{2}}}}$$
a) $$\displaystyle{g{{\left({f{{\left(-{2}\right)}}}\right)}}}$$
Find and simplify in expression for the idicated composite functions. State the domain using interval notation.
$$\displaystyle{f{{\left({x}\right)}}}={3}{x}-{1}$$
$$\displaystyle{g{{\left({x}\right)}}}=\frac{{1}}{{{x}+{3}}}$$
Find $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}$$
The number of electric scooters e that a factory can produce per day is a function of the number of hours h it operates and is given by $$\displaystyle{e}{\left({h}\right)}={290}{h},{0}\le{h}\le{10}.$$
The daily cost c to manufacture e electric scooters is given by the function $$\displaystyle{c}{\left({e}\right)}={0.05}{e}^{{2}}+{65}{e}+{1000}.$$
(a) Find $$\displaystyle{\left({c}\circ{e}\right)}{\left({h}\right)}.$$
(b) Evaluate $$\displaystyle{\left({c}\circ{e}\right)}{\left({13}\right)}.$$
Let f(x) = $$\displaystyle{4}{x}^{{2}}–{6}$$ and $$\displaystyle{g{{\left({x}\right)}}}={x}–{2}.$$
(a) Find the composite function $$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}$$ and simplify. Show work.
(b) Find $$\displaystyle{\left({f}\circ{g}\right)}{\left(−{1}\right)}$$. Show work.