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Chapter 2: Problem 69

Determine whether each relation defines \(y\) as a function of \(x\). \(y=\frac{2}{x-4}\)

Short Answer

Expert verified
Yes, \(y = \frac{2}{x-4}\) defines \(y\) as a function of \(x\).

Step by step solution

01

- Understand the relation

The given relation is a rational function: \[ y = \frac{2}{x-4} \]
02

- Identify and Consider Domain Restrictions

This function will be undefined where the denominator is equal to zero. Find the value of \(x\) that makes the denominator zero: \[ x - 4 = 0 \] Solving for \(x\), we get: \[ x = 4 \] So, the function is undefined at \(x = 4\).
03

- Determine if \(y\) is a Function of \(x\)

A relation defines \(y\) as a function of \(x\) if every \(x\) in the domain maps to exactly one \(y\). For all \(x eq 4\), substituting \(x\) into the equation \( y = \frac{2}{x-4} \) gives exactly one value of \(y\). Therefore, the relation defines \(y\) as a function of \(x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain Restrictions
Domain restrictions in a function refer to the values of the variable for which the function is not defined. For rational functions, these values occur where the denominator is zero.

In the function \( y = \frac{2}{x-4} \), the denominator becomes zero when \(x = 4\). This is because substituting \(x = 4\) into the denominator, we get \[x - 4 = 0 \]. Solving this equation, we find \[ x = 4 \].

Therefore, the function \( y = \frac{2}{x-4} \) is not defined at \(x = 4\), and \(x = 4\) is a domain restriction. Excluding \(x = 4\), the function is defined for all other real numbers.
Undefined Values
Undefined values in a rational function occur at points where the function's denominator equals zero.

In the given function, \( y = \frac{2}{x-4} \), the denominator \(x - 4\) becomes zero when \(x = 4\). This makes the function \y\) undefined at this point because division by zero is not possible in mathematics.

Therefore, at \(x = 4\), the value of \ y\ becomes undefined, as substituting \( x = 4\ in \( y = \frac{2}{x-4} \) results in division by zero.
Function Definition
A relation defines \y\ as a function of \x\ if every \x\ in the domain maps to exactly one \y\. This means each input has a single, unique output.

For the function \( y = \frac{2}{x-4} \), we need to check if every \x\ (except \(x = 4\)) maps to one unique \y\.

When substituting any \ x eq 4\ into \( y = \frac{2}{x-4} \), there is always one unique value of \ y\. Therefore, this relation is indeed a function because for every input (excluding \(x = 4\)), there is a unique output.

Thus, the given relation \( y = \frac{2}{x-4} \) defines \ y\ as a function of \ x\.

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