Math CentralQuandaries & Queries


Question from marylie, a student:

what is the domain and range of f(x)= -sq.rootof x ^2-4??tnx


Let me do a slightly different question

What are the domain and range of f(x) = √(x4 - 81)?


It's the square root here that can cause a problem. You can't take the square root of a negative number so you must have

x4 - 81 ≥ 0

This means that

x4 ≥ 81

The fourth root of 81 is 3 so either x ≥ 3 or x ≤ -3. Hence the domain of f(x) is all real numbers x so that x ≥ 3 or x ≤ -3.


The question here is for what numbers y is it possible to find a number x so that y = √(x4 - 81)? The operator √ always returns a non-negative number so y must be non-negative but given a non-negative number y can you find an x so that y = √(x4 - 81)?

If you square both sides you get y2 = x4 - 81 and hence x4 = y2 + 81. The right side is positive so I can take the fourth root to get x = (y2 + 81)1/4.


You give me a non-negative number y and I take x = (y2 + 81)1/4. Then

√(x4 - 81) = √)[(y2 + 81)1/4]4 - 81) = √(y2 + 81 - 81) = √y2 and since y was non-negative √y2 = y.

Thus the range of f(x) is all non-negative numbers.

I hope this helps,


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