91Ó°ÊÓ

The function we are examining is a fourth root function, specifically written as \( f(x) = \sqrt[4]{x-9} \). This type of function involves finding the fourth root (or the square root, applied twice) of the expression inside the radical.
The fourth root function has its unique characteristics:

• It only produces real numbers when its argument (the value inside the root) is non-negative, meaning it must be zero or greater.
• Unlike square root functions, fourth root functions are defined for a broader range of inputs because the root taken is even, which allows negative outputs.

To grasp the domain of \( f(x) = \sqrt[4]{x-9} \), we must ensure that the expression \( x-9 \) is non-negative, ensuring the existence of real fourth roots.

To determine the valid input values for our fourth root function, we analyze the inequality derived from \( x-9 \geq 0 \). This means we need to find the x-values that make \( x-9 \) non-negative (greater than or equal to zero).
Inequalities like this are solved by isolating the variable:

• Start with the inequality \( x-9 \geq 0 \).
• Add 9 to both sides to isolate x: \Rightarrow \( x \geq 9 \).

This tells us that any x-value of 9 or higher will satisfy the inequality, ensuring the expression inside the fourth root is non-negative.
This step-by-step approach is essential to understanding how the domain of our function is determined.

In the context of our function \( f(x) = \sqrt[4]{x-9} \), we focus on the domain, which is concerned with the set of real numbers that make the function valid. Real numbers include all rational numbers (fractions and integers) and irrational numbers (numbers that cannot be expressed as fractions, like \( \sqrt{2} \)).
As established from our inequality \( x \geq 9 \), the domain of our function is specifically all real numbers equal to or greater than 9. Therefore, for any real number x starting from 9 and extending to infinity, the fourth root function \( f(x) \) will produce a valid result.
This understanding of real numbers and their role in defining the domain helps clarify when and how our function can be used. It emphasizes the continuous nature of real numbers and the importance of ensuring a non-negative argument in root functions.