Without the exponent of 1/2 for a moment...
x^12 - x^9 + x^4 - x + 1
is a 12th degree polynomial.
That is an even degree.
So the graph of
y = x^12 - x^9 + x^4 - x + 1
will have end behavior that is the same on the left and right.
Since the coefficient of the 12th degree term is an understood positive 1, the end behavior will be up on the left and up on the right. It MAY have up to 11 wiggles in between, but those won't affect the domain.
It is simply a polynomial function, so the domain is all real numbers.
You have not asked about the range. Finding the range would be a bit more complicated. The maximum would be infinite, since the graph is up on the left and up on the right. However, finding the minimum range would be a bit tougher. It would involve calculus: take the derivative and then set the derivative equal to zero to find the x-values of the local minima points. Plug those in to the derivative to find the y-values. The smallest y-value obtained would be the small end of the range.
Now, let's add in the 1/2 power.
A 1/2 power means a square root (radical with index of 2).
An exponent of 1/2 puts the whole expression under a square root.
The only limits on the domain would be x-values that would cause the expression under the square root to be negative.
Since the polynomial is of even degree and up on the left and right, it is possible that none of the wiggles drop below the x-axis.
How can we check?
Considering this largest terms...
The x^12 term would be always positive, and very, very large (even if x is negative!)
The x^9 term would not be nearly so large, so subtracting it (when x is negative) would not yield a negative result under the radical.
So we can conclude that no value for x will produce a negative value under the square root.
Therefore no value for x will be ruled out of the domain.
So, the domain is all real numbers!
Domain: -∞ > x > ∞
Or in interval notation:
D: (-∞,∞)
For reference, the graph is...
(Wyzant's limit on characters would not allow for the graph.)
It continues to extend outward to the left and right, ever so gradually, but extending nonetheless all the way to negative infinity on the left and positive infinity on the right.
