Function: f(x) = 2x² + 3
• There is no restriction for x, so the domain of f is R (all real numbers).
f is a quadratic function, with a = 2 > 0. So, f has a minimum value.
f(x) = 2x² + 3 ———> a = 2, b = 0, c = 3
Finding the discriminant:
Δ = b² – 4ac
Δ = 0² – 4 · 2 · 3
Δ = – 24
The minimum value of f is the y-coordinate of the vertex:
f_min = y_V
Δ
f_min = – ———
4a
(– 24)
f_min = – ————
4 · 2
24
f_min = ———
8
f_min = 3
• So the range of f is
{y ∈ R: y ≥ f_min}
{y ∈ R: y ≥ 3}
or using the interval notation,
[3, +∞[.
I hope this helps. =)
Tags: quadratic function domain range minimum value vertex algebra
