Math  /  Algebra

Question29. [-/0.35 Points] DETAILS MY NOTES SCOLALG7 3.7.043. O/100 Submissions Used
Find the domain of the given function. (Enter your answer using interval notation.) h(x)=x414h(x)=\sqrt[4]{x^{4}-1} \square Need Help? Read It Submit Answer

Studdy Solution

STEP 1

What is this asking? We need to find all the allowed xx values for the function h(x)=x414h(x) = \sqrt[4]{x^4 - 1}. Watch out! We can't take the fourth root of a negative number!

STEP 2

1. Analyze the function
2. Find the domain

STEP 3

Alright, so we've got this function h(x)=x414h(x) = \sqrt[4]{x^4 - 1}, and we're looking for its **domain**.
The domain is just all the *valid* inputs, all the xx values we can plug in without breaking any math rules.

STEP 4

The key here is that fourth root.
We know we can't take the fourth root (or any even root) of a negative number.
So, whatever is *inside* that radical, the x41x^4 - 1, has to be greater than or equal to zero.
Let's write that down!
x410 x^4 - 1 \ge 0

STEP 5

Now, let's solve this inequality!
We want to isolate xx, so let's add one to both sides:
x41 x^4 \ge 1

STEP 6

Since we have xx to the fourth power, we can take the fourth root of both sides.
Remember, when taking an even root of both sides of an inequality, the inequality sign stays the same.
x4414 \sqrt[4]{x^4} \ge \sqrt[4]{1} x1 |x| \ge 1

STEP 7

Remember that x|x| means the absolute value of xx.
So, this inequality means that xx can be any number whose *absolute value* is greater than or equal to **one**.
That means xx can be **one** or bigger, *or* xx can be **negative one** or smaller.

STEP 8

Let's write that in **interval notation**.
We've got two separate intervals: from negative infinity up to and including negative one, and from one up to infinity.
We can write that as:
(,1][1,) (-\infty, -1] \cup [1, \infty)

STEP 9

The domain of the function h(x)=x414h(x) = \sqrt[4]{x^4 - 1} is (,1][1,)(-\infty, -1] \cup [1, \infty).

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