For this case we have the following function:
[tex]y = 2 (x + 2) ^ 2 - 3
[/tex]
Since the leading coefficient is positive, then the parabola opens upwards.
Since the function has no restriction for any value of x, we have then that the domain is all real numbers.
To find the range, we must find the vertex of the parabola.
For this, we derive:
[tex]y '= 4 (x + 2)
[/tex]
We equal zero and clear x:
[tex]4 (x + 2) = 0
x + 2 = 0
x = -2[/tex]
We evaluate x = -2 in the original function:
[tex]y = 2 (-2 + 2) ^ 2 - 3
y = 2 (0) ^ 2 - 3
y = 0 - 3
y = - 3[/tex]
The vertice of the parabola is:
[tex](x, y) = (- 2, -3)
[/tex]
Therefore, the range of the function is:
[-3, ∞)
The minimum value is:
[tex]y = -3
[/tex]
Answer:
B. minimum value: -3
domain: all real numbers
range: all real numbers ≥ -3
