# Find and simplify in expression for the idicated composite functions. State the domain using interval notation. f(x)=3x-1 g(x)=1/(x+3) Find ([email protected])(x)

Question
Composite functions
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 $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}$$

2021-02-26
Obtain the composite function, $$\displaystyle{g}\circ{f{{\left({x}\right)}}}$$
$$\displaystyle{g}\circ{f{{\left({X}\right)}}}={g{{\left({f{{\left({x}\right)}}}\right)}}}$$
$$\displaystyle=\frac{{1}}{{{\left({f{{\left({x}\right)}}}\right)}+{3}}}$$
$$\displaystyle=\frac{{1}}{{{\left({3}{x}-{1}\right)}+{3}}}$$
$$\displaystyle=\frac{{1}}{{{3}{x}+{2}}},{x}\ne-\frac{{2}}{{3}}$$
Here, the simplified expression for the composite function is
$$\displaystyle{g}\circ{f{{\left({x}\right)}}}=\frac{{1}}{{{3}{x}+{2}}},{x}\ne-\frac{{2}}{{3}}$$
The composite function defined for all values of x except at $$\displaystyle{x}=-\frac{{2}}{{3}}$$ that is defined all values on the left and right of $$\displaystyle{x}=-\frac{{2}}{{3}}$$ but not at $$\displaystyle{x}=-\frac{{2}}{{3}}$$, thus the domain in the interval notation becomes $$\displaystyle{\left(-\infty,-\frac{{2}}{{3}}\right)}\cup{\left(\frac{{2}}{{3}},\infty\right)}$$

### 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
f(x) = 3x + 1
g(x) = −x
Decide whether the composite functions, $$\displaystyle{f}\circ{g}$$ and $$\displaystyle{g}\circ{f}$$, are equal to x.
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{9}$$
$$\displaystyle{g{{\left({x}\right)}}}={\sqrt[{{3}}]{{{x}-{9}}}}$$
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
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 $$\displaystyle{g}\circ{f}$$?
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)
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.
$$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{x}}-{1}$$
$$\displaystyle{g{{\left({x}\right)}}}=\sqrt{{{1}-{x}^{{2}}}}$$
a) $$\displaystyle{g{{\left({f{{\left(-{2}\right)}}}\right)}}}$$
$$\displaystyle{f{{\left({x}\right)}}}={71}{e}^{{{0.2}{x}}}$$