Problem 14 - Elementary Algebra

Question

For the following problems, find the domain of each of the rational expressions. $$ \frac{x+10}{x+4} $$

Step-by-Step Solution

Answer

Question: Find the domain of the rational expression \(\frac{x+10}{x+4}\). Answer: The domain of the rational expression is all real numbers except x = -4. In interval notation, the domain is \((-\infty,-4)\cup(-4,\infty)\).

Step 1: Identify the denominator

Identify the denominator of the rational expression, which is (x+4) in this case.

Step 2: Find the values that make the denominator zero

Set the denominator (x+4) equal to zero and solve for x: $$ x + 4 = 0 $$ Subtract 4 from both sides: $$ x = -4 $$

Step 3: Exclude values that make the denominator zero

Now that we found the value of x that makes the denominator equal to zero, we know that x cannot be equal to -4. This is because division by zero is undefined.

Step 4: Write the domain

The domain of the rational expression is all real numbers except x= -4. In interval notation, the domain can be written as: $$ (-\infty,-4)\cup(-4,\infty) $$

Key Concepts

Domain of a Function

The domain of a function is a fundamental concept in mathematics. It represents all possible input values (typically represented as \(x\)) for which the function is defined and gives a valid output. In simple terms, it's the set of 'allowable' numbers you can plug into the function without causing any math problems.

For rational expressions, such as \(\frac{x+10}{x+4}\), finding the domain involves identifying any restrictions. These restrictions often occur if there's a risk of dividing by zero, which is undefined in mathematics.

To find the domain:

• Identify any restrictions, such as zero denominators.
• Exclude these values from the domain.

Therefore, checking the denominator is crucial in determining the domain of a rational expression.

Denominator

The denominator in a fraction is the bottom part of the fraction, representing the number of parts into which the whole is divided. In the case of rational expressions, such as \(\frac{x+10}{x+4}\), the expression \(x+4\) in the denominator plays a critical role.

Having a zero in the denominator makes the fraction undefined because division by zero does not yield a finite number or meaning.

Here are steps to understand the importance of the denominator:

• Always identify the denominator before proceeding with the solving or simplification of a rational expression.
• Set the denominator equal to zero to find any restrictions.
• Remember that the domain must exclude these values to remain valid.

The denominator dictates the values that need to be omitted from the function's domain due to its role in fractional calculations.

Division by Zero

Division by zero is a special topic in mathematics as it is undefined. Attempting to divide any number by zero leads to a situation without a clear or usable result, which is why it is typically avoided in mathematical operations.

When dealing with rational expressions like \(\frac{x+10}{x+4}\), division by zero occurs when the denominator equals zero. For this example, setting \(x+4 = 0\) tells us that \(x = -4\) would lead to division by zero.

To avoid this, it is essential:

• Identify when division by zero would occur in your calculations.
• Exclude these values from the domain of the function.

By correctly identifying and avoiding division by zero, you ensure the rational expression remains valid and defined over its domain.