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.
Solution:
Classify the Function
Polynomial Function
A polynomial function is a function of the form where n is a non-negative integer {0, 1, 2, 3, 4, …} and the coefficients |
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Rational Function
A rational function is a function of the form where |
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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 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.
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Solve the inequality. This inequality is a linear inequality and can be solved by isolating the variable on one side. | ![]() ![]() |
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. | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
The function values where are defined for
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In set builder notation, the domain is .
In interval notation, the domain is (]