 # Let f(x) = 4x^2–6 and g(x)=x–2.(a) Find the composite function (f@g)(x) and simplify. Show work.(b) Find (f@g)(−1). Show work. Chesley 2020-12-25 Answered

Let f(x) = $4{x}^{2}–6$ and $g\left(x\right)=x–2.$
(a) Find the composite function $\left(f\circ g\right)\left(x\right)$ and simplify. Show work.
(b) Find $\left(f\circ g\right)\left(-1\right)$. Show work.

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