Will give abrainlest

which of the following accurately shows the range of the function defined below?

[3,infinty)
[2,infinty)
[3,6]
(-infinty, infinty)

We are going to study the range of the function part by part.

 For x <0:
 The function is:
 f (x) = 3
 The range of the function is 3.

 For 0 <= x <2
 The function is:
 f (x) = x ^ 2 + 2
 When x = 0 we have:
 f (0) = (0) ^ 2 + 2 = 2
 When x = 2 we have:
 f (2) = (2) ^ 2 + 2 = 6
 The range of the function is [2, 6)

 For x> = 2
 The function is:
 f (x) = (1/2) x + 5
 When x = 2 we have:
 f (2) = (1/2) (2) + 5 = 6
 The range of the function is [6, inf)

 The range of the complete function is:
 [3] U [2, 6) U [6, inf)
 Equivalently:
 [2, INFINTY)

  
 [2, INFINTY)