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.

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
asked 2020-12-25
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.

Answers (1)

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
0

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