Problem 111 - Precalculus Essentials

Question

How do you determine if an equation in \(x\) and \(y\) defines \(y\) as a function of \(x ?\)

Step-by-Step Solution

Answer

An equation in \(x\) and \(y\) defines \(y\) as a function of \(x\) if and only if each value of \(x\) corresponds to exactly one value of \(y\). One way to test this is to analyze the equation and check that for each \(x\), there is only one possible \(y\). If unsure, plotting the equation and ensuring it passes the vertical line test can help.

Step 1: Understanding the concept of a function

In Mathematics, a function can be defined as a special relation where every input (\(x\)) gives a unique output (\(y\)). This means, for a given value of \(x\), there should only be one corresponding value of \(y\).

Step 2: Analyzing the given equation

The given equation should be inspected to see if every input \(x\) produces exactly one output \(y\). If there are instances where a single value of \(x\) leads to multiple distinct possible values of \(y\), then the equation does not define \(y\) as a function of \(x\).

Step 3: Plotting the given equation (Optional)

If the equation appears complicated to analyze by inspection, plotting it on a graph can be useful. For an equation in \(x\) and \(y\) to define a function, it must pass the vertical line test. If a vertical line drawn at any value of \(x\) on the graph intersects the curve at more than one point, then the equation does not define \(y\) as a function of \(x\).