To determine if the equation [tex]\( y = \sqrt[13]{x} \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to understand the definition of a function. A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.
Here’s how we can analyze the given equation [tex]\( y = \sqrt[13]{x} \)[/tex]:
1. Understanding the equation:
- The equation [tex]\( y = \sqrt[13]{x} \)[/tex] expresses [tex]\( y \)[/tex] as the 13th root of [tex]\( x \)[/tex].
- The 13th root function, like any odd root function, is defined for all real numbers [tex]\( x \)[/tex].
2. Unique Output for Each Input:
- For each real number [tex]\( x \)[/tex], the 13th root function provides exactly one real number output [tex]\( y \)[/tex].
- This is because odd roots, such as the 1st, 3rd, 5th, and in this case, the 13th root, always yield a single real output for any real input.
3. Function Definition:
- Since for any given input [tex]\( x \)[/tex], there is exactly one output [tex]\( y \)[/tex], [tex]\( y \)[/tex] is uniquely determined by [tex]\( x \)[/tex].
- This satisfies the definition of a function, as each input [tex]\( x \)[/tex] has a single corresponding output [tex]\( y \)[/tex].
Given this understanding, we can conclude that the equation [tex]\( y = \sqrt[13]{x} \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]. Therefore, the correct answer to the question is:
B. Yes, because for any input [tex]\( x \)[/tex], the equation yields only one output [tex]\( y \)[/tex].
