Chapter 9.2, Problem 90AYU

Explain why the vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.

To determine

To explain: The vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.

Answer to Problem 90AYU

The vertical line test is a method that is used to determine whether a given relation is a function or not. The approach is rather simple. Draw a vertical line cutting through the n graph of the relation, and then observe the points of intersection. The vertical line test supports the definition of a function. That is, every $x\text{-value}$ of a function must be paired to a single $y\text{-value}$.

For example: Polar equation $r = 0.1\theta$ represents function in polar plane. Generally the function like $y = 0.1x$ in the Cartesian plane can be stated as a linear function and satisfies the vertical line test. However, in polar plane, this graph is spiral. This violates the vertical line test and is thus not a function. The vertical line test is not really helpful in the polar system.

Explanation of Solution

The vertical line test is a method that is used to determine whether a given relation is a function or not. The approach is rather simple. Draw a vertical line cutting through the n graph of the relation, and then observe the points of intersection. The vertical line test supports the definition of a function. That is, every $x\text{-value}$ of a function must be paired to a single $y\text{-value}$.

For example: Polar equation $r = 0.1\theta$ represents function in polar plane. Generally the function like $y = 0.1x$ in the Cartesian plane can be stated as a linear function and satisfies the vertical line test. However, in polar plane, this graph is spiral. This violates the vertical line test and is thus not a function. The vertical line test is not really helpful in the polar system.