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- llafferLv 71 month agoFavorite Answer
You are asked to find the domain of that function.

The domain is a list of all possible inputs into the function that will result in a real output.

Since your function is the fourth root, any negative value inside will result in a complex number so we know what's inside of it must be zero or positive. We can now create an inequlality based on this info that we can solve:

x² - 9x ≥ 0

Factor an x:

x(x - 9) ≥ 0

We know if either of the two factors are zero then the product is zero which will satisfy the inequality. Those roots are:

x = 0 and 9

We can then use those roots as pivots to find out which range(s) will count.

if x > 9 we have positive times positive which is positive, in our solution set.

if 0 < x < 9 we have positive times negative which is negative, not in our solution set.

if x < 0 we have negative times negative which is postive, in our solution set.

And again, the points 0 and 9 are valid solutions to our derived inequality so that solution is:

x ≤ 0 and x ≥ 9

since those values will result in a non-negative value under the fourth root, the result will be a real number. This makes the above solution also the result of the domain for this function.