What is the range of the function f (x)=2x^2+3? i already found the domain

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