Answer
(a.)$(f\circ g)(x) =20x^2-11$.
(b.)$(g\circ f)(x) =80x^2-120x+43$.
(c.)$(f\circ g)(2) =69$.
Work Step by Step
The given functions are
$f(x)=4x-3$ and $g(x)=5x^2-2$.
(a) $(f\circ g)(x) = f(g(x))$
Replace $x$ with $g(x)$ in the function $f(x)$.
$f(g(x))=4(g(x))-3$
Plug value of $g(x)$ in the right hand side.
$f(g(x))=4(5x^2-2)-3$
$f(g(x))=20x^2-8-3$
$f(g(x))=20x^2-11$.
Hence, $(f\circ g)(x) =20x^2-11$.
(b) $(g\circ f)(x) = g(f(x))$
Replace $x$ with $f(x)$ in the function $g(x)$.
$g(f(x))=5(f(x))^2-2$
Plug value of $f(x)$ in the right hand side.
$g(f(x))=5(4x-3)^2-2$
$g(f(x))=5(16x^2+9-24x)-2$
$g(f(x))=80x^2+45-120x-2$
$g(f(x))=80x^2-120x+43$.
Hence, $(g\circ f)(x) =80x^2-120x+43$.
(c) Replace $x$ with $2$ in the part (a) solution.
$(f\circ g)(2) =20(2)^2-11$
$(f\circ g)(2) =20(4)-11$
$(f\circ g)(2) =80-11$
$(f\circ g)(2) =69$.
