Example of a Piecewise Relation

Oct 13, 2025 by wbrum.org 95 views

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function is a way of assigning each input to exactly one output. In this article, we will explore which piecewise relation defines a function.

What is a Piecewise Relation?

A piecewise relation is a relation that is defined by multiple sub-relations, each of which is defined on a specific subset of the domain. In other words, a piecewise relation is a relation that is composed of multiple relations, each of which is defined on a specific part of the domain.

Example of a Piecewise Relation

Let's consider the following piecewise relation:

y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned}\right.

This piecewise relation is defined on the domain $(-\infty, -2) \cup [-2, 4] \cup [4, \infty)$ and is composed of three sub-relations:

$y = x^2$ for $x < -2$
$y = 0$ for $-2 \leq x \leq 4$
$y = -x^2$ for $x \geq 4$

What Makes a Piecewise Relation a Function?

A piecewise relation is a function if and only if each sub-relation is a function. In other words, a piecewise relation is a function if and only if each sub-relation assigns each input to exactly one output.

Analyzing the Piecewise Relation A

Let's analyze the piecewise relation A:

y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned}\right.

We can see that each sub-relation is a function:

$y = x^2$ is a function for $x < -2$ because each input $x$ is assigned to exactly one output $x^2$ .
$y = 0$ is a function for $-2 \leq x \leq 4$ because each input $x$ is assigned to exactly one output $0$ .
$y = -x^2$ is a function for $x \geq 4$ because each input $x$ is assigned to exactly one output $-x^2$ .

Therefore, the piecewise relation A is a function.

Analyzing the Piecewise Relation B

Let's analyze the piecewise relation B:

y=\left\{\begin{array}{cl} x^2, & \quad x < 0 \\ 0, & \quad x = 0 \\ x^2, & \quad x > 0 \end{array}\right.

We can see that the sub-relation $y = x^2$ for $x < 0$ is a function, but the sub-relation $y = x^2$ for $x > 0$ is not a function because each input $x$ is assigned to two outputs $x^2$ and $x^2$ .

Therefore, the piecewise relation B is not a function.

Conclusion

In conclusion, a piecewise relation is a function if and only if each sub-relation is a function. We analyzed two piecewise relations, A and B, and determined that A is a function while B is not a function.

Key Takeaways

A piecewise relation is a relation that is defined by multiple sub-relations, each of which is defined on a specific subset of the domain.
A piecewise relation is a function if and only if each sub-relation is a function.
Each sub-relation must assign each input to exactly one output for the piecewise relation to be a function.

Discussion

In our previous article, we explored what makes a piecewise relation a function. In this article, we will answer some frequently asked questions about piecewise relations and functions.

Q: What is a piecewise relation?

A piecewise relation is a relation that is defined by multiple sub-relations, each of which is defined on a specific subset of the domain.

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

A piecewise relation is a relation that is defined by multiple sub-relations, while a function is a relation that assigns each input to exactly one output. A piecewise relation is a function if and only if each sub-relation is a function.

Q: How do I determine if a piecewise relation is a function?

To determine if a piecewise relation is a function, you need to check if each sub-relation is a function. This means that each sub-relation must assign each input to exactly one output.

Q: What are some examples of piecewise relations?

Here are some examples of piecewise relations:

$y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned}\right.$
$y=\left\{\begin{array}{cl} x^2, & \quad x < 0 \\ 0, & \quad x = 0 \\ x^2, & \quad x > 0 \end{array}\right.$

Q: Can a piecewise relation have multiple sub-relations?

Yes, a piecewise relation can have multiple sub-relations. For example:

y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad 4 < x \leq 6 \\ x^3, & \quad x > 6 \end{aligned}\right.

Q: Can a piecewise relation have a sub-relation that is not a function?

Yes, a piecewise relation can have a sub-relation that is not a function. For example:

y=\left\{\begin{array}{cl} x^2, & \quad x < 0 \\ 0, & \quad x = 0 \\ x^2, & \quad x > 0 \end{array}\right.

In this example, the sub-relation $y = x^2$ for $x > 0$ is not a function because each input $x$ is assigned to two outputs $x^2$ and $x^2$ .

Q: Can a piecewise relation have a sub-relation that is a constant function?

Yes, a piecewise relation can have a sub-relation that is a constant function. For example:

y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned}\right.

In this example, the sub-relation $y = 0$ for $-2 \leq x \leq 4$ is a constant function because each input $x$ is assigned to the same output $0$ .

Q: Can a piecewise relation have a sub-relation that is a linear function?

Yes, a piecewise relation can have a sub-relation that is a linear function. For example:

y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned}\right.

In this example, the sub-relation $y = x^2$ for $x < -2$ is a quadratic function, but it can be rewritten as a linear function in terms of $x^2$ . Similarly, the sub-relation $y = -x^2$ for $x \geq 4$ is a quadratic function, but it can be rewritten as a linear function in terms of $-x^2$ .

Q: Can a piecewise relation have a sub-relation that is a polynomial function?

Yes, a piecewise relation can have a sub-relation that is a polynomial function. For example:

y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned}\right.

In this example, the sub-relation $y = x^2$ for $x < -2$ is a quadratic function, which is a polynomial function of degree 2. Similarly, the sub-relation $y = -x^2$ for $x \geq 4$ is a quadratic function, which is a polynomial function of degree 2.

Q: Can a piecewise relation have a sub-relation that is a rational function?

Yes, a piecewise relation can have a sub-relation that is a rational function. For example:

y=\left\{\begin{aligned} x^2, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned}\right.

Q: Can a piecewise relation have a sub-relation that is a trigonometric function?

Yes, a piecewise relation can have a sub-relation that is a trigonometric function. For example:

y=\left\{\begin{aligned} \sin(x), & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -\sin(x), & \quad x \geq 4 \end{aligned}\right.

In this example, the sub-relation $y = \sin(x)$ for $x < -2$ is a trigonometric function, which is a periodic function with period $2\pi$ . Similarly, the sub-relation $y = -\sin(x)$ for $x \geq 4$ is a trigonometric function, which is a periodic function with period $2\pi$ .

Q: Can a piecewise relation have a sub-relation that is an exponential function?

Yes, a piecewise relation can have a sub-relation that is an exponential function. For example:

y=\left\{\begin{aligned} e^x, & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -e^x, & \quad x \geq 4 \end{aligned}\right.

In this example, the sub-relation $y = e^x$ for $x < -2$ is an exponential function, which is a function of the form $y = a^x$ where $a$ is a positive constant. Similarly, the sub-relation $y = -e^x$ for $x \geq 4$ is an exponential function, which is a function of the form $y = a^x$ where $a$ is a positive constant.

Q: Can a piecewise relation have a sub-relation that is a logarithmic function?

Yes, a piecewise relation can have a sub-relation that is a logarithmic function. For example:

y=\left\{\begin{aligned} \log(x), & \quad x < -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -\log(x), & \quad x \geq 4 \end{aligned}\right.

In this example, the sub-relation $y = \log(x)$ for $x < -2$ is a logarithmic function, which is a function of the form $y = \log(x)$ where $x$ is a positive real number. Similarly, the sub-relation $y = -\log(x)$ for $x \geq 4$ is a logarithmic function, which is a function of the form $y = \log(x)$ where $x$ is a positive real number.

Conclusion

In conclusion, a piecewise relation can have multiple sub-relations, each of which can be a function of a different type, such as a polynomial function, a rational function, a trigonometric function, an exponential function, or a logarithmic function. We hope this article has provided a clear understanding of what makes a piecewise relation a function and how to determine if a piecewise relation is a function.

Key Takeaways

A piecewise relation is a relation that is defined by multiple sub-relations, each of which is defined on a specific subset of the