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6.4.1: Resources and Key Concepts
Opening the book…
Resources
Key Concepts
Definitions
Radical Function: A function that is defined by a radical expression.
Theorems
Properties of \(\sqrt[n]{a}\) (Regarding Real Numbers):
When \(n\) is an even number and \(a \ge 0\), then \(\sqrt[n]{a}\) is a real number.
When \(n\) is an even number and \(a < 0\), then \(\sqrt[n]{a}\) is not a real number.
When \(n\) is an odd number, \(\sqrt[n]{a}\) is a real number for all values of \(a\).
Domain of a Radical Function:
When the index of the radical is even, the domain of the radical function consists of all real numbers for which the radicand is greater than or equal to zero.
When the index of the radical is odd, the domain of a radical function consists of all real numbers (assuming the radicand itself is defined for all real numbers).
Common Mistakes
Ignoring Domain Restrictions for Even-Indexed Radicals: Attempting to evaluate an even-indexed radical function for an input that makes the radicand negative, or failing to correctly identify the domain by setting the radicand \(\ge 0\).
Forgetting Denominator Restrictions within Radicands: When the radicand of an even-indexed root is a rational expression, not only must the entire rational expression be \(\ge 0\), but the denominator of that rational expression also cannot be zero.