91Ó°ÊÓ

A fourth root function, like the one in the exercise, is a special type of root function where we find the number that, when multiplied by itself three more times, equals the given radicand. In simpler terms, the fourth root of a number is a value that, when used in four equal factors, gives the original number back. The function \(g(x) = (x-6)^{1/4}\) characterizes this because the exponent \(\frac{1}{4}\) signifies the fourth root.

Root functions can sometimes create complex numbers if not handled correctly, particularly because of even roots. This is why it's essential to keep the radicand non-negative to ensure the results remain within the realm of real numbers. The concept of a fourth root function ensures that the function output is real and calculable.

Solving inequalities is a process very similar to solving equations, with slight differences in handling the inequality symbol. Here, we need to solve the inequality \(x-6 \geq 0\) to find where the radicand (\(x-6\)) of the fourth root function is valid.

**Steps for Solving the Inequality:**

• Initially, we understand that for any root with an even degree, such as a fourth root, the expression inside the root must be \(\geq 0\) for the result to be a real number.
• We start with the inequality \(x-6 \geq 0\).
• Add 6 to both sides to isolate \(x\), resulting in \(x \geq 6\).

This process tells us that \(x\) must be equal to or greater than 6, ensuring the radicand is non-negative, therefore giving real number results.

The domain of a function is the complete set of possible values of the independent variable that will produce a valid output. For real number domains, we only consider inputs for which the function yields real results.

In the case of the function \(g(x) = (x-6)^{1/4}\), we determined that only values of \(x\), where \(x \geq 6\), are acceptable. This is found through our inequality solving process, ensuring the radicand remains non-negative.

**Expressing the Domain:**

• The domain is all real numbers \(x\) such that \(x \geq 6\).
• In interval notation, this domain is written as \([6, \infty)\).

Such notation clearly communicates that \(x\) starts at 6 and goes up to positive infinity, ensuring the function stays real and operational within calculus and real-world applications.