Problem 16 - Elementary Algebra

Question

Find the domain of the equations. Assume that the independent variable is the variable that appears in the expression to the right of the equal sign. $$ m=\frac{-16}{h} $$

Step-by-Step Solution

Answer

Answer: The domain of the given equation is all real values of h except for h = 0, which can be written as \(h \in \mathbb{R} - \{0\}\).

Step 1: Identify the restriction on the independent variable

In the given equation, \(m = \frac{-16}{h}\), a possible restriction is that h cannot be equal to 0, since division by zero is undefined.

Step 2: Find the domain

Based on the identified restriction, the domain of the equation is all real values of h except for h = 0, which can be written as: \(h \in \mathbb{R} - \{0\}\)

Key Concepts

Independent Variable

When dealing with equations like \( m = \frac{-16}{h} \), one of the key components to identify is the independent variable. An independent variable is a value that you can change freely in an equation or function, and it often determines the outcome of a dependent variable. In this case, the independent variable is \( h \), which appears on the right side of the equation after the equal sign.
This means that any change in \( h \) will directly affect the value of \( m \).

• The independent variable can usually take a variety of values.
• It sets the stage for how we determine the domain of a function.

Understanding which variable is independent is crucial for solving equations correctly, as it guides you in identifying any necessary restrictions.

Division by Zero

Division by zero is a mathematical operation that is undefined in standard arithmetic. In essence, if you have a division expressed as \( \frac{a}{b} \), where \( b \) equals zero, the result cannot be determined or expressed within the real number system. This is because you cannot divide a number into zero parts.
In the equation \( m = \frac{-16}{h} \), if \( h \) were to become zero, it would result in a division by zero, rendering the expression undefinable. Here’s why this is significant:

• Division by zero is an invalid operation and must be avoided.
• When \( h = 0 \), the expression \( \frac{-16}{h} \) does not produce a valid number.
• This leads us to put \( h eq 0 \) as a condition for our domain.

By understanding this principle, you can better identify restrictions on the independent variables that will affect the domain of your function.

Undefined Expressions

Undefined expressions occur when a mathematical operation does not result in a real number. This usually happens in cases like division by zero, taking the square root of a negative number (in the real number system), or other invalid operations.
For the function \( m = \frac{-16}{h} \), substituting \( h = 0 \) would result in an undefined expression due to division by zero. Thus:

• If an expression is undefined for a particular value of the independent variable, such values are excluded from the domain.
• In this example, the expression is undefined when \( h = 0 \), so \( h = 0 \) must be excluded.
• Expressions should be evaluated to ensure they are defined for all values within the intended domain.

The process of identifying and addressing undefined expressions is crucial when finding the domain for any function or equation. This helps to prevent mathematical errors and ensures valid, accurate computations.