Question

Writing to Learn The vertical line test to determine whether a curve is the graph of a function states: If every vertical line in the \(x y\) -plane intersects a given curve in at most one point, then the curve is the graph of a function. Explain why this is true.

Short answer

The vertical line test corresponds to the definition of a function concerning having a single output for a specific input. If every vertical line intersects a curve at most at one point, it shows that each input (x-value) has one output (y-value), thus respecting the function's definition. If a vertical line intersects the curve at more than one point, multiple outputs are present for the same input, which contradicts the function definition.

Step-by-step solution

Define a function

In mathematics, a function can be defined as a relationship between a set of inputs and a set of permissible outputs where each input is related to exactly one output.

Connect function definition with vertical line test

The vertical line test is a visual way to determine if a curve is a graph of a function or not. The test is used by drawing a line parallel to the y-axis for any chosen value of x. If the vertical line intersects the graph more than once, it means for that value of x there are multiple outputs, which contradicts the definition of a function.

Conclude

Therefore, if every vertical line (for each value of x) intersects a given curve at most at one point, then the curve represents a function, respecting the definition. If any line intersects the curve at more than one point, then the curve cannot represent a function because it contradicts the function's definition.

Key concepts

Function Definition

Understanding the concept of a function is crucial for exploring various aspects of mathematics. Simply put, a function is like a machine that takes an input, processes it, and gives out a unique output. Formally, in the realm of mathematics, a function is a special relationship where each input is connected to exactly one output. Imagine you have a box of apples and you want to assign a unique sticker to each apple. If you can do this without any apple sharing the same sticker, then you have a function.

However, the concept goes deeper. This definition implies that for each value from the set of inputs (often called the domain), there is a corresponding and unique value in the set of outputs (known as the range). The representation not only ensures that every input has an output but also stipulates that no single input can produce two different outputs.

Graph of a Function

Moving from the abstract to the visual, the graph of a function provides us with a picture of what's happening in a function. You could think of it as a map or a snapshot that shows all the possible connections between inputs and their corresponding outputs. When functions are graphed, they can take numerous forms, such as lines, curves, and shapes, on a coordinate plane.

A coordinate plane has two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). The point where these axes intersect is called the origin. When graphing, the input value is shown on the x-axis, whereas the output value is shown on the y-axis.

Plotting the Function

To graph a function, you take a set of input values, compute the outputs, and plot these pairs of numbers as points. When these points are connected, if they form a continuous curve or line without any breaks, they are likely representing a graph of a function.

Function's Input and Output Relationship

Diving into the heart of functions, the function's input and output relationship is akin to a dance between numbers. Each input gets its moment to pair with an output in such a way that it's exclusive; no two inputs share the same output. The beauty of this relationship is its consistency and predictability.

In every function, we can select any input from the domain, and we can determine its output in the range. This is an ordered affair, and this orderliness is what makes a graph represent a function.

The Vertical Line Test

A popular method to confirm whether a graphed curve truly follows this ordered relationship is the vertical line test. This test involves imagining (or drawing) vertical lines (parallel to the y-axis) across the graph. If at any point a vertical line touches the graph in more than one spot, the dance is off, and you're looking at two outputs for one input—which means it is not a proper function. This test is a quick visual confirmation of the one-to-one relationship required of a function.