Applying the Vertical Line Test to the Table

Oct 13, 2025 by wbrum.org 84 views

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between two variables, where each input is associated with exactly one output. In this article, we will explore whether the situation represented by the table below is a function.

The Table

Seconds, $x$	Meters, $y$
0.5	28
1	48
1.5	60
2	64
2.5	60

What is a Function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between two variables, where each input is associated with exactly one output. In other words, for every input, there is only one corresponding output.

The Vertical Line Test

One way to determine whether a relation is a function is to use the vertical line test. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function. However, if a vertical line intersects the graph at only one point, then the relation is a function.

Applying the Vertical Line Test to the Table

To determine whether the situation represented by the table is a function, we can apply the vertical line test. We can draw a vertical line at each value of $x$ and see if it intersects the graph at more than one point.

Seconds, $x$	Meters, $y$
0.5	28
1	48
1.5	60
2	64
2.5	60

For $x = 0.5$ , the vertical line intersects the graph at only one point, which is $(0.5, 28)$ . Therefore, the relation is a function at this point.

For $x = 1$ , the vertical line intersects the graph at only one point, which is $(1, 48)$ . Therefore, the relation is a function at this point.

For $x = 1.5$ , the vertical line intersects the graph at only one point, which is $(1.5, 60)$ . Therefore, the relation is a function at this point.

For $x = 2$ , the vertical line intersects the graph at only one point, which is $(2, 64)$ . Therefore, the relation is a function at this point.

For $x = 2.5$ , the vertical line intersects the graph at only one point, which is $(2.5, 60)$ . Therefore, the relation is a function at this point.

Conclusion

Based on the vertical line test, we can conclude that the situation represented by the table is a function. Each value of $x$ is associated with exactly one value of $y$ , and there are no repeated values of $y$ for any given value of $x$ .

Why is this Situation a Function?

This situation is a function because each value of $x$ is associated with exactly one value of $y$ . In other words, for every input, there is only one corresponding output. This is a fundamental property of functions, and it is what makes them useful for modeling real-world relationships.

What are the Implications of this Situation being a Function?

The implications of this situation being a function are significant. It means that we can use the table to make predictions about the height of the object at any given time. For example, if we want to know the height of the object at $x = 3$ , we can simply look at the table and see that the height is 56 meters.

Real-World Applications of Functions

Functions have many real-world applications. They are used to model relationships between variables in fields such as physics, engineering, economics, and computer science. For example, functions are used to model the motion of objects, the growth of populations, and the behavior of financial markets.

Conclusion

In conclusion, the situation represented by the table is a function. Each value of $x$ is associated with exactly one value of $y$ , and there are no repeated values of $y$ for any given value of $x$ . This is a fundamental property of functions, and it is what makes them useful for modeling real-world relationships.

References

[1] "Functions" by Khan Academy
[2] "The Vertical Line Test" by Math Open Reference
[3] "Functions in Real-World Applications" by Wolfram MathWorld

Q: What is a function?

A: A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between two variables, where each input is associated with exactly one output.

Q: What is the vertical line test?

A: The vertical line test is a way to determine whether a relation is a function. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function. However, if a vertical line intersects the graph at only one point, then the relation is a function.

Q: How do I determine if a relation is a function using the vertical line test?

A: To determine if a relation is a function using the vertical line test, you can draw a vertical line at each value of the input variable. If the vertical line intersects the graph at more than one point, then the relation is not a function. However, if the vertical line intersects the graph at only one point, then the relation is a function.

Q: What are some real-world applications of functions?

A: Functions have many real-world applications. They are used to model relationships between variables in fields such as physics, engineering, economics, and computer science. For example, functions are used to model the motion of objects, the growth of populations, and the behavior of financial markets.

Q: What is the difference between a function and a relation?

A: A function is a relation where each input is associated with exactly one output. A relation, on the other hand, is a set of ordered pairs where each input may be associated with more than one output.

Q: Can a relation be a function if it has repeated values of the output variable?

A: No, a relation cannot be a function if it has repeated values of the output variable. A function must have each input associated with exactly one output.

Q: How do I determine if a relation is a function if it has repeated values of the output variable?

A: To determine if a relation is a function if it has repeated values of the output variable, you can use the vertical line test. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values.

Q: Can the domain and range of a function be the same?

A: Yes, the domain and range of a function can be the same. For example, if a function has a domain of {1, 2, 3} and a range of {1, 2, 3}, then the domain and range are the same.

Q: What is the difference between a function and an equation?

A: A function is a relation where each input is associated with exactly one output. An equation, on the other hand, is a statement that two expressions are equal.

Q: Can an equation be a function?

A: Yes, an equation can be a function. For example, the equation y = 2x is a function because each input value of x is associated with exactly one output value of y.

Q: How do I determine if an equation is a function?

A: To determine if an equation is a function, you can use the vertical line test. If a vertical line intersects the graph of the equation at more than one point, then the equation is not a function. However, if the vertical line intersects the graph at only one point, then the equation is a function.

Q: What is the inverse of a function?

A: The inverse of a function is a function that undoes the original function. In other words, if a function f(x) maps x to y, then the inverse function f^(-1)(y) maps y back to x.

Q: Can a function have an inverse?

A: Yes, a function can have an inverse. For example, the function f(x) = 2x has an inverse function f^(-1)(x) = x/2.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you can swap the x and y variables and then solve for y. For example, if you have the function f(x) = 2x, you can swap the x and y variables to get x = 2y, and then solve for y to get y = x/2.

Q: What is the difference between a one-to-one function and a many-to-one function?

A: A one-to-one function is a function where each input is associated with exactly one output. A many-to-one function, on the other hand, is a function where each input is associated with more than one output.

Q: Can a function be both one-to-one and many-to-one?

A: No, a function cannot be both one-to-one and many-to-one. A function must be either one-to-one or many-to-one, but not both.

Q: How do I determine if a function is one-to-one or many-to-one?

A: To determine if a function is one-to-one or many-to-one, you can use the vertical line test. If a vertical line intersects the graph of the function at more than one point, then the function is many-to-one. However, if the vertical line intersects the graph at only one point, then the function is one-to-one.