Explanation:
the domain is the interval or set of all valid x- values.
the range is the interval or set of all valid y-values.
so, the domain here is (-infinity .. +infinity).
as there is clearly nothing hindering us to go all the way to the left or all the way to the right to pick an x-value for the function.
the range here is (-infinity .. +3].
as the y-value can "dive" down as deep as we want, but it cannot get larger than +3.
please consider the choice of brackets. "(", ")" means excluding the specified interval ends ("infinity" is a concept, not a value, so it can never be included).
"[", "]" means including the specified interval ends.
the function is clearly a parabola.
y = ax² + bx + c
the parabola opens downwards, so a is negative.
for x = 0, we get f(x) = 3, so c = +3
for y = 0, we get x = ±2
so,
0 = a2² + b2 + 3
and
0 = a(-2)² + b(-2) + 3
4a + 2b = -3
4a - 2b = -3
--------------------
8a 0 = -6
a = -6/8 = -3/4
4a + 2b = -3
4(-3/4) + 2b = -3
-3 + 2b = -3
2b = 0
b = 0
the function is
f(x) = (-3/4)x² + 3