Combined Domain Restrictions

Oct 13, 2025 by leolavanderia.com.br 79 views

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with rational expressions, it's essential to consider the domain restrictions to ensure that the expression is valid and well-defined. In this article, we will explore the domain restrictions for the given expression:

\frac{14m - 14}{56m^2 - 56m} \cdot \frac{8m^2 + 48m}{2m - 2}

Domain Restrictions

The given expression is a product of two rational expressions. To find the domain restrictions, we need to consider the values of $m$ that make each rational expression undefined. Let's analyze each expression separately:

Expression 1: $\frac{14m - 14}{56m^2 - 56m}$

The expression is undefined when the denominator is equal to zero. To find the values of $m$ that make the denominator zero, we need to solve the equation:

56m^2 - 56m = 0

Factoring out the common term $56m$ , we get:

56m(m - 1) = 0

This gives us two possible values for $m$ :

m = 0 \quad \text{or} \quad m = 1

Therefore, the expression $\frac{14m - 14}{56m^2 - 56m}$ is undefined when $m = 0$ or $m = 1$ .

Expression 2: $\frac{8m^2 + 48m}{2m - 2}$

The expression is undefined when the denominator is equal to zero. To find the values of $m$ that make the denominator zero, we need to solve the equation:

2m - 2 = 0

Solving for $m$ , we get:

m = 1

Therefore, the expression $\frac{8m^2 + 48m}{2m - 2}$ is undefined when $m = 1$ .

Combined Domain Restrictions

Combining the domain restrictions for both expressions, we get:

$m \neq 1$
$m \neq -1$
$m \neq 0$
$m \neq -6$

These are the domain restrictions for the given expression.

Discussion

The domain restrictions for the given expression are essential to ensure that the expression is valid and well-defined. When dealing with rational expressions, it's crucial to consider the domain restrictions to avoid undefined values.

In this case, the domain restrictions are:

$m \neq 1$
$m \neq -1$
$m \neq 0$
$m \neq -6$

These restrictions ensure that the expression is defined for all values of $m$ except for the specified values.

Conclusion

In conclusion, the domain restrictions for the given expression are:

$m \neq 1$
$m \neq -1$
$m \neq 0$
$m \neq -6$

These restrictions are essential to ensure that the expression is valid and well-defined. When dealing with rational expressions, it's crucial to consider the domain restrictions to avoid undefined values.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart

Keywords

Domain restrictions
Rational expressions
Algebra
Calculus
Mathematics
Domain Restrictions for the Given Expression: Q&A =====================================================

Introduction

In our previous article, we explored the domain restrictions for the given expression:

\frac{14m - 14}{56m^2 - 56m} \cdot \frac{8m^2 + 48m}{2m - 2}

We found that the domain restrictions are:

$m \neq 1$
$m \neq -1$
$m \neq 0$
$m \neq -6$

In this article, we will answer some frequently asked questions (FAQs) related to the domain restrictions for the given expression.

Q&A

Q: What is the significance of domain restrictions in mathematics?

A: Domain restrictions are essential in mathematics to ensure that a function or expression is valid and well-defined. When dealing with rational expressions, it's crucial to consider the domain restrictions to avoid undefined values.

Q: Why are the values $m = 1$ , $m = -1$ , $m = 0$ , and $m = -6$ excluded from the domain?

A: These values are excluded from the domain because they make the denominator of the expression zero, which is undefined.

Q: Can you explain why $m = 1$ is excluded from the domain?

A: Yes, $m = 1$ is excluded from the domain because it makes the denominator of the first expression zero. Specifically, when $m = 1$ , the denominator $56m^2 - 56m$ becomes zero.

Q: What happens if we try to evaluate the expression at $m = 0$ ?

A: If we try to evaluate the expression at $m = 0$ , we will get an undefined value. This is because the denominator of the first expression becomes zero when $m = 0$ .

Q: Can you provide an example of a value of $m$ that is not excluded from the domain?

A: Yes, a value of $m$ that is not excluded from the domain is $m = 2$ . This value is not equal to any of the excluded values ( $m = 1$ , $m = -1$ , $m = 0$ , or $m = -6$ ), so it is a valid input for the expression.

Q: How do we determine the domain restrictions for a rational expression?

A: To determine the domain restrictions for a rational expression, we need to find the values of the variable that make the denominator zero. These values are excluded from the domain.

Q: Can you explain the concept of a rational expression?

A: A rational expression is a fraction of two polynomials. It is a type of algebraic expression that involves division.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a type of number that can be expressed as the ratio of two integers. A rational expression, on the other hand, is a type of algebraic expression that involves division.