# Let f(x) = 4x^2–6 and g(x)=x–2. (a) Find the composite function ([email protected])(x) and simplify. Show work. (b) Find ([email protected])(−1). Show work.

Question
Composite functions
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.

2020-12-26
(a) The composite function is,
$$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}$$
$$\displaystyle={f{{\left({x}-{2}\right)}}}$$
$$\displaystyle={4}{\left({x}-{2}\right)}^{{2}}-{6}$$ [replacing x by x-2 in f(x)]
$$\displaystyle={4}{\left({x}^{{2}}-{4}{x}+{4}\right)}-{6}$$
$$\displaystyle={4}{x}^{{2}}-{16}{x}+{16}-{6}$$
$$\displaystyle{4}{x}^{{2}}-{16}{x}+{10}$$
b) Replacing x by -1 in the composite function, we get
$$\displaystyle{\left({f}\circ{g}\right)}{\left(-{1}\right)}={4}{\left(-{1}\right)}^{{2}}-{16}{\left(-{1}\right)}+{10}$$
=4+16+10
=30

### Relevant Questions

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
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$$\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{9}$$
$$\displaystyle{g{{\left({x}\right)}}}={\sqrt[{{3}}]{{{x}-{9}}}}$$
Given
$$\displaystyle{f{{\left({x}\right)}}}={2}-{x}{2},{g{{\left({x}\right)}}}=\sqrt{{{x}+{2}}}$$
$$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}},{g{{\left({x}\right)}}}=\sqrt{{{1}-{x}}}$$
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(b) domain and
(c) range of each.
$$\displaystyle{f{{\left({x}\right)}}}=\frac{{5}}{{{x}+{9}}}$$
$$\displaystyle{f{{\left({x}\right)}}}={3}{x}-{1}$$
$$\displaystyle{g{{\left({x}\right)}}}=\frac{{1}}{{{x}+{3}}}$$
Find $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}$$