Understanding the Original Relation

Oct 13, 2025 by thesweetdesigns.com 97 views

Understanding the Original Relation

The given relation is $x = -4|y| + 2$ . This equation represents a linear relationship between the variables $x$ and $y$ , where the absolute value of $y$ is involved. The absolute value function $|y|$ indicates that the relation is not symmetric about the origin, and its graph will have a V-shape.

Inverse of the Relation

To find the inverse of the relation, we need to interchange the variables $x$ and $y$ and then solve for $y$ . The inverse relation is obtained by swapping the $x$ and $y$ values in the original equation.

Interchanging Variables

Interchanging the variables $x$ and $y$ in the original equation $x = -4|y| + 2$ gives us:

y = -4|x| + 2

Solving for $y$

Now, we need to solve for $y$ in the equation $y = -4|x| + 2$ . This equation represents a linear relationship between the variables $x$ and $y$ , where the absolute value of $x$ is involved.

Finding the Inverse Relation

To find the inverse relation, we need to consider the cases when $x$ is positive and when $x$ is negative.

Case 1: $x \geq 0$

When $x \geq 0$ , the absolute value of $x$ is equal to $x$ . Substituting this into the equation $y = -4|x| + 2$ gives us:

y = -4x + 2

Case 2: $x < 0$

When $x < 0$ , the absolute value of $x$ is equal to $-x$ . Substituting this into the equation $y = -4|x| + 2$ gives us:

y = 4x + 2

Finding Four Points Contained in the Inverse Relation

To find four points contained in the inverse relation, we need to find the values of $x$ and $y$ that satisfy the equation $y = -4|x| + 2$ .

Point 1: $(0, 2)$

When $x = 0$ , the equation $y = -4|x| + 2$ becomes:

y = -4(0) + 2

y = 2

So, the point $(0, 2)$ is contained in the inverse relation.

Point 2: $(1, -2)$

When $x = 1$ , the equation $y = -4|x| + 2$ becomes:

y = -4(1) + 2

y = -2

So, the point $(1, -2)$ is contained in the inverse relation.

Point 3: $(-1, 6)$

When $x = -1$ , the equation $y = -4|x| + 2$ becomes:

y = -4(-1) + 2

y = 6

So, the point $(-1, 6)$ is contained in the inverse relation.

Point 4: $(-2, 10)$

When $x = -2$ , the equation $y = -4|x| + 2$ becomes:

y = -4(-2) + 2

y = 10

So, the point $(-2, 10)$ is contained in the inverse relation.

Conclusion

In this article, we found four points contained in the inverse of the relation $x = -4|y| + 2$ . The points are $(0, 2)$ , $(1, -2)$ , $(-1, 6)$ , and $(-2, 10)$ . These points satisfy the equation $y = -4|x| + 2$ , which represents the inverse relation.

References

[1] Khan Academy. (n.d.). Absolute Value. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f6/x2f6f7/x2f6f8
[2] Math Open Reference. (n.d.). Inverse of a Function. Retrieved from https://www.mathopenref.com/inversefunction.html

Keywords

Inverse relation
Absolute value
Linear relationship
Function
Inverse function
Points contained in the inverse relation

Understanding the Inverse Relation

The inverse relation is a concept in mathematics that is used to describe the relationship between two variables. In the previous article, we discussed the inverse relation of the equation $x = -4|y| + 2$ . In this article, we will answer some frequently asked questions about the inverse relation and absolute value.

Q&A

Q: What is the inverse relation?

A: The inverse relation is a concept in mathematics that is used to describe the relationship between two variables. It is obtained by interchanging the variables $x$ and $y$ in the original equation.

Q: How do you find the inverse relation?

A: To find the inverse relation, you need to interchange the variables $x$ and $y$ in the original equation and then solve for $y$ .

Q: What is the absolute value function?

A: The absolute value function is a mathematical function that returns the absolute value of a number. It is denoted by $|x|$ and is defined as:

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Q: How do you use the absolute value function in the inverse relation?

A: In the inverse relation, the absolute value function is used to describe the relationship between the variables $x$ and $y$ . When $x$ is positive, the absolute value function is equal to $x$ . When $x$ is negative, the absolute value function is equal to $-x$ .

Q: What are some examples of inverse relations?

A: Some examples of inverse relations include:

$y = -4|x| + 2$
$y = 2|x| - 3$
$y = -3|x| + 1$

Q: How do you graph the inverse relation?

A: To graph the inverse relation, you need to plot the points that satisfy the equation and then connect them with a line.

Q: What are some real-world applications of the inverse relation?

A: The inverse relation has many real-world applications, including:

Physics: The inverse relation is used to describe the relationship between the variables in a physical system.
Engineering: The inverse relation is used to design and optimize systems.
Economics: The inverse relation is used to model the relationship between variables in an economic system.

Conclusion

In this article, we answered some frequently asked questions about the inverse relation and absolute value. We discussed the concept of the inverse relation, how to find it, and how to use the absolute value function in it. We also provided some examples of inverse relations and discussed their real-world applications.

References

[1] Khan Academy. (n.d.). Absolute Value. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f6/x2f6f7/x2f6f8
[2] Math Open Reference. (n.d.). Inverse of a Function. Retrieved from https://www.mathopenref.com/inversefunction.html

Keywords

Inverse relation
Absolute value
Linear relationship
Function
Inverse function
Real-world applications
Physics
Engineering
Economics