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How can I find the domain of (4 - x^4)^(1/4) ?

I am a student brushing up on my algebra basics for a Calculus course (and placement exam) via Coursera Calculus I hosted by OSU. This question was on one of the practice quizzes: What is the domain of f(x) = the fourth root of (4 - x^4)?

I found the answer, [-sqrt(2), sqrt(2)] by pattern matching and verifying this guess in Wolfram. However, I need to understand its mathematical basis (since Wolfram didn't provide a step-by-step for this solution). Here's the logic I undertook, and where, exactly, my reasoning falls through:

  1. To find the domain of f(x) = the fourth root of (4 - x^4), I must factor the expression inside the square root. This is a difference of two squares problem. Factorization lead me to (-x^2+2)(x^2+2). I could go a step further and factor out a -1, but I get the same result either way with fewer steps.

  2. I now need to set these two expressions as separate inequalities and solve. In the case of (-x^2+2), I arrive at the solution x <= sqrt(2) with no problem. However, the second inequality poses a logical problem:

          x^2+2 >= 0
    
          x^2 >= -2 
    

Here's where I fail to make the logical leap to the answer I'm seeking. At this point in the course, I've been lead to believe that it is not possible to take the square root of a negative integer (though I have been introduced to imaginary numbers in high school.) I suppose this might mean multiplying both sides of the inequality by -1 to reach -x^2 >= 2, but how is it possible to bypass the conventional order of operations to arrive at x >= -sqrt(2)?

I really appreciate any help anyone can offer. My math skills are a little rusty and I've been known to under/overthink problems like these.

Best regards.

3 Replies
Posted 9 years ago

The domain may be obtained as follows

Reduce[(4 - x^4) >= 0, x]

-Sqrt[2] <= x <= Sqrt[2]
POSTED BY: Imran Khan

Another approach, which uses the definition of domain, is

In[3]:= Reduce[res == (4 - x^4)^(1/4), {x, res}, Reals]

Out[3]= -Sqrt[2] <= x <= Sqrt[2] && res == (4 - x^4)^(1/4)
POSTED BY: Frank Kampas

This site is about Wolfram Technologies. And while you do mention it, your actual question is not about Wolfram Technologies but about doing math by hand. You should not be asking this questions on this forum, but find another dedicated place, for example HERE. Both Wolfram|Alpha and Wolfram Language can give you the final answer though.

FunctionDomain[(4 - x^4)^(1/4), x]
NumberLinePlot[FunctionDomain[(4 - x^4)^(1/4), x], {x, -2, 2}]
Plot[(4 - x^4)^(1/4), {x, -2, 2}]

enter image description here

POSTED BY: Sam Carrettie
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