SOLUTION: The domain of the function q(x) = x^4 + 4x^2 + 4 is [0,\infty). What is the range? No matter what, I can't seem to get the right answer.

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Question 1169641: The domain of the function q(x) = x^4 + 4x^2 + 4 is [0,\infty). What is the range?
No matter what, I can't seem to get the right answer.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
sketch it:
+graph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+10%2C+x%5E4+%2B+4x%5E2+%2B+4%29+
y-axis intersected at 4-> point (0,4)
if given that the domain of the function q%28x%29+=+x%5E4+%2B+4x%5E2+%2B+4 is
[0,infinity)

then the range is: { q element R : q%3E=4 }
in interval notation: [4,infinity)

Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


x%5E4%2B4x%5E2%2B4+=+%28x%5E2%2B2%29%5E2

Since the original expression is the square of a real number, the original expression is always greater than or equal to 0.

x%5E2%2B2 is always 2 or greater, because x%5E2 alone is always 0 or greater.

Then, since the minimum value of x%5E2%2B2 is 2, the minimum value of x%5E4%2B4x%5E2%2B4+=+%28x%5E2%2B2%29%5E2 is 2%5E2=4

ANSWER: The range of x%5E4%2B4x%5E2%2B4 is [4,infinity).

An even easier path to that answer is to note that both x%5E4 and x%5E2 are always 0 or greater (with the minimum value of both at x=0); that means the minimum value of x%5E4%2B4x%5E2%2B4 is 0+0+4 = 4.