Finding Domain: Radical/Root Function (Even Root)

Example: Classify the function as a polynomial function, rational function, or root function, and then find the domain. Write the domain interval notation and set builder notation.

f(x) = root{6}{2-x}

Solution:

Classify the Function

Formal Definition	Practical Way of Identifying
Polynomial Function A polynomial function is a function of the form $f(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_1 x+a_0$ where n is a non-negative integer {0, 1, 2, 3, 4, …} and the coefficients $a_n , a_{n-1,...,a_1,a_0$ are from the real numbers.	Look for the variables to be in the numerator. (If there is no fraction at all, they are in the numerator.) The variable should not be inside a radical or absolute value. The powers or exponents on the variables should be whole numbers. Whole numbers come from this list {0, 1, 2, 3, 4, …}
Rational Function A rational function is a function of the form where and are polynomial functions and is not equal to zero.	The numerator and denominator are polynomials. Most functions with variables in the denominator are considered Rational Functions but there are exceptions.
Root Function (even index) A root function is a function of the form where n is an even positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an even number. {2, 4, 6, 8, …} The square root is an even index although the index is not written.
Root Function (odd index) A root function is a function of the form where n is an odd positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an odd number. {3, 5, 7, 9, …} The cube root is an odd index.

Since the function f(x) = root{6}{2-x} has a radical and the index is even. This function is a root function.

Find the Domain of a Root Function (Even Index)

Taking the even root of a negative number results in a complex or imaginary number. Since we are interested in real function values, we would like the expression inside the radical to be non-negative ( zero or positive) The root function is defined for any value of the variable where the expression under the radical is non-negative (zero or positive). Find these values by creating an inequality to solve. The inequality is the expression under the radical greater than or equal to zero.

Set the expression under the radical greater than or equal to zero.	0
Solve the inequality. This inequality is a linear inequality and can be solved by isolating the variable on one side.	0
Solve by isolating the variable. Start by subtracting 2 on both sides.
Isolate the variable. Continue by dividing both sides by -1. Be sure to reverse the inequality symbol since you are dividing both sides by a negative.