Problem 111 - College Algebra 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

To determine if an equation in \(x\) and \(y\) defines \(y\) as a function of \(x\), check if for each value of \(x\), there is exactly one corresponding value of \(y\). This can be done by solving for \(y\) in the equation for different values of \(x\) or by using the vertical line test on the graph of the equation.

Step 1: Understand the concept of a function

By definition, a function from set \(A\) to set \(B\) is a rule that assigns to each element of set \(A\) exactly one element of set \(B\). Hence, if an equation defines \(y\) as a function of \(x\), to every value of \(x\), there corresponds exactly one value of \(y\).

Step 2: Test the equation

Pick different values for \(x\), substitute them into the equation and solve for \(y\). If for every value of \(x\), there is exactly one corresponding value of \(y\), then the equation defines \(y\) as a function of \(x\).

Step 3: Verify with the vertical line test

Plot the equation on a graph. Draw vertical lines at various points along the graph. If any of these vertical lines intersect the graph at more than one point, the equation does not define \(y\) as a function of \(x\), because it means there are multiple values of \(y\) for the same value of \(x\). If every vertical line intersects the graph at exactly one point, then \(y\) is a function of \(x\) because there is exactly one value of \(y\) for each value of \(x\).