Question

Consider the function g, domain R, defined by f(x)=2x^2+4x-1.

What is the codomain of the g function?

A: ]-∞,-3]
B: [-1,+∞[
C: [-3,+∞[
D: ]-∞,-1]

So, the answer to this question is option C, but I don't know why. Can anyone explain it to me?

Answer 1

see explanation

Explanation:

the codomain are the values of f(x) from the y- coordinate of the vertex upwards.

given a quadratic in standard form

y = ax² + bx + c ( a ≠ 0 )

then the x- coordinate of the vertex is

`x_(V)` = -
`(b)/(2a)`

f(x) = 2x² + 4x - 1 ← is in standard form

with a = 2 and b = 4 , then

`x_(V)` = -
`(4)/(4)` = - 1

substitute x = - 1 into f(x) for y- coordinate of vertex

f(- 1) = 2(- 1)² + 4(- 1) - 1 = 2 - 4 - 1 = - 3

vertex = (- 1, - 3 )

since the coefficient of the x² term > 0 then the graph opens up U

then codomain is [ - 3, ∞ )