Hi – I've got this question about finding the domain of a function, and I got the answer, but the method I used is quite different from the explanation provided. My question is: Is my method flawed in any way? I'm still uncertain when it comes to taking square roots of squares, let alone fourth roots.
Here's the original question:
What's the domain of
$$f(x)= \sqrt[4]{4-x^4}$$
The correct answer is
$$[-\sqrt{2}, \sqrt{2}]$$
and I'm fine with that.
The explanation given involves using the difference of squares and figuring out when they both have the same (positive) sign. Fine, I understand that. It wouldn't have occurred to me to use that.
I worked it out a little differently, and wonder if my reasoning was right:
$$4-x^4\geq 0$$
$$x^4\leq 4$$
$$\sqrt[4]{x^4}\leq\sqrt[4]{4}$$
$$\sqrt {\sqrt{x^4}}\leq\sqrt {\sqrt{4}}$$
$$\sqrt{x^2}\leq\sqrt{2}$$
$$|x|\leq\sqrt{2}$$
$$[-\sqrt{2}, \sqrt{2}]$$
I'm a little shaky on step 4/5: should there be absolute value signs around or positive signs in front of, the two reductions?
Thanks!
Best Answer
Your method is correct. For steps 4-5, there are 'implied' absolute value signs in your solution: $x^2$ is always positive (for real $x$), and you correctly only wrote down the positive solution to $\sqrt{4}$.