Consider The Following Relation:${ X = -5|y| + 4 }$Step 2 Of 2: Find The Domain And Range Of The Inverse. Express Your Answer In Interval Notation.

Mar 9, 2025 by ADMIN 149 views

**Understanding the Inverse Relation**

In mathematics, an inverse relation is a fundamental concept that plays a crucial role in various branches of mathematics, including algebra, geometry, and calculus. The inverse of a relation is a new relation that is obtained by swapping the input and output values of the original relation. In this article, we will explore the concept of inverse relations and find the domain and range of the inverse of the given relation $x = -5|y| + 4$ .

Step 1: Understanding the Original Relation

The original relation is given by the equation $x = -5|y| + 4$ . This equation represents a linear function that takes an input value $y$ and produces an output value $x$ . The absolute value function $|y|$ is used to ensure that the output value $x$ is always non-negative.

Step 2: Finding the Inverse Relation

To find the inverse relation, we need to swap the input and output values of the original relation. This means that we need to solve the equation for $y$ in terms of $x$ . We can start by isolating the absolute value term $|y|$ on one side of the equation.

x = -5|y| + 4
|y| = (x - 4) / -5

Since the absolute value function $|y|$ is always non-negative, we can remove the absolute value sign and rewrite the equation as:

y = (x - 4) / -5

However, we need to consider two cases: when $y$ is non-negative and when $y$ is negative. When $y$ is non-negative, the equation becomes:

y = (x - 4) / -5

When $y$ is negative, the equation becomes:

y = -(x - 4) / 5

Step 3: Finding the Domain and Range of the Inverse

Now that we have found the inverse relation, we need to find the domain and range of the inverse. The domain of a relation is the set of all possible input values, while the range is the set of all possible output values.

Domain of the Inverse

The domain of the inverse relation is the set of all possible input values $x$ that produce a valid output value $y$ . Since the original relation is a linear function, the domain of the inverse relation is all real numbers.

Range of the Inverse

The range of the inverse relation is the set of all possible output values $y$ that correspond to a valid input value $x$ . Since the original relation is a linear function, the range of the inverse relation is also all real numbers.

However, we need to consider the two cases: when $y$ is non-negative and when $y$ is negative. When $y$ is non-negative, the range of the inverse relation is:

y ≥ 0

When $y$ is negative, the range of the inverse relation is:

y < 0

Conclusion

In conclusion, the inverse relation of the given relation $x = -5|y| + 4$ is $y = (x - 4) / -5$ when $y$ is non-negative and $y = -(x - 4) / 5$ when $y$ is negative. The domain of the inverse relation is all real numbers, while the range is all real numbers. However, we need to consider the two cases: when $y$ is non-negative and when $y$ is negative.

Final Answer

The final answer is:

In this article, we will answer some frequently asked questions about inverse relations, including the concept of inverse relations, finding the inverse of a relation, and the domain and range of the inverse.

Q: What is an inverse relation?

A: An inverse relation is a new relation that is obtained by swapping the input and output values of the original relation. In other words, if we have a relation $x = f(y)$ , then the inverse relation is $y = f^{-1}(x)$ .

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

A: To find the inverse of a relation, we need to swap the input and output values of the original relation. This means that we need to solve the equation for $y$ in terms of $x$ . We can start by isolating the variable $y$ on one side of the equation.

Q: What is the difference between the original relation and the inverse relation?

A: The original relation and the inverse relation are two different relations that are obtained by swapping the input and output values of the original relation. The original relation is the function $f(y)$ , while the inverse relation is the function $f^{-1}(x)$ .

Q: How do I find the domain and range of the inverse relation?

A: To find the domain and range of the inverse relation, we need to consider the two cases: when $y$ is non-negative and when $y$ is negative. When $y$ is non-negative, the domain of the inverse relation is all real numbers, while the range is $y \geq 0$ . When $y$ is negative, the domain of the inverse relation is all real numbers, while the range is $y < 0$ .

Q: What is the significance of the inverse relation?

A: The inverse relation is significant because it allows us to solve equations and inequalities that involve the original relation. By finding the inverse relation, we can solve for the input value $x$ in terms of the output value $y$ .

Q: Can I use the inverse relation to solve equations and inequalities?

A: Yes, you can use the inverse relation to solve equations and inequalities that involve the original relation. By substituting the inverse relation into the equation or inequality, you can solve for the input value $x$ in terms of the output value $y$ .

Q: What are some common applications of inverse relations?

A: Inverse relations have many common applications in mathematics, including:

Solving equations and inequalities
Finding the domain and range of a relation
Graphing relations
Calculating the inverse of a matrix

Q: Can I use inverse relations to solve real-world problems?

A: Yes, you can use inverse relations to solve real-world problems that involve relations. By applying the concept of inverse relations, you can solve problems that involve functions, graphs, and matrices.

Conclusion

In conclusion, inverse relations are an important concept in mathematics that allows us to solve equations and inequalities, find the domain and range of a relation, and graph relations. By understanding the concept of inverse relations, you can apply it to solve real-world problems that involve relations.

Final Answer

The final answer is:

Inverse relation: A new relation that is obtained by swapping the input and output values of the original relation.
Domain of the inverse relation: All real numbers.
Range of the inverse relation: $y \geq 0$ when $y$ is non-negative and $y < 0$ when $y$ is negative.
Significance of the inverse relation: Allows us to solve equations and inequalities that involve the original relation.
Applications of inverse relations: Solving equations and inequalities, finding the domain and range of a relation, graphing relations, calculating the inverse of a matrix.