# Decide whether the composite functions, [email protected] and [email protected], are equal to x. f(x)=x^3+9 g(x)=root(3)(x-9)

Question
Composite functions
Decide whether the composite functions, $$\displaystyle{f}\circ{g}$$ and $$\displaystyle{g}\circ{f}$$, are equal to x.
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{9}$$
$$\displaystyle{g{{\left({x}\right)}}}={\sqrt[{{3}}]{{{x}-{9}}}}$$

2020-10-28
To find the composite function we need to plug-in function in place of the variable. Mathematically,
$$\displaystyle{f}\circ{g}={f{{\left({g{{\left({x}\right)}}}\right)}}}$$
$$\displaystyle{g}\circ{f}={g{{\left({f{{\left({x}\right)}}}\right)}}}$$
Therefore, the composite functions are,
$$\displaystyle{f}\circ{g}={f{{\left({\sqrt[{{3}}]{{{x}-{9}}}}\right)}}}$$
$$\displaystyle={\sqrt[{{3}}]{{{x}-{9}}}}+{9}$$
$$\displaystyle={x}-{9}+{9}$$
$$\displaystyle={x}$$
$$\displaystyle{g}\circ{f}={g{{\left({f{{\left({x}\right)}}}\right)}}}$$
$$\displaystyle={\sqrt[{{3}}]{{{\left({x}^{{3}}+{9}\right)}-{9}}}}$$
$$\displaystyle={\sqrt[{{3}}]{{{x}^{{3}}}}}$$
=x
Hence, both composite functions are equal to zero.

### Relevant Questions

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 $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}$$
For f(x)=6/x and g(x)=6/x, find the following functions. a) ([email protected])(x) b) ([email protected])(x) c) ([email protected])(7) d) ([email protected])(7)
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.
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.
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 $$\displaystyle{g}\circ{f}$$?
$$\displaystyle{f{{\left({x}\right)}}}=\frac{{5}}{{{x}+{9}}}$$
$$\displaystyle{f{{\left({x}\right)}}}={71}{e}^{{{0.2}{x}}}$$
Let $$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}+{9}{x}$$ and $$\displaystyle{g{{\left({x}\right)}}}={9}{x}+{8}$$ perform the composition or operation indicated below $$\displaystyle{\left({f}\circ{g}\right)}{\left({5}\right)}$$ simplify the answer