Answer
(a.) Maximum.
(b.) At $x=1$, maximum value is $1$.
(c.) Domain $=(-\infty,\infty)$.
Range $=(-\infty,1]$.
Work Step by Step
The given function is a quadratic function:
$f(x)=-4x^2+8x-3$
The standard form of the quadratic function is
$f(x)=ax^2+bx+c$
$a=-4,b=8$ and $c=-3$
(a.)
Because $a<0$, the function has a maximum value.
(b.)
$x-$coordinate at which maximum value occurs is
$x=-\frac{b}{2a}$.
Substitute all values.
$x=-\frac{(8)}{2(-4)}$.
Simplify.
$x=1$.
Substitute the value of $x$ into the given function.
$f(1)=-4(1)^2+8(1)-3$
Simplify.
$f(1)=-4+8-3$
$f(1)=1$
(c.)
Domain is all possible input values.
Domain $=(-\infty,\infty)$.
Range is all possible output values.
maximum value is $1$.
Range $=(-\infty,1]$.
