Problem 1 - Algebra and Trigonometry

Question

find all numbers that must be excluded from the domain of each rational expression. $$ \frac{7}{x-3} $$

Step-by-Step Solution

Answer

The number that must be excluded from the domain of the rational expression \( \frac{7}{x-3} \) is 3.

Step 1: Identify the expression

The function to analyze is \( \frac{7}{x-3} \).

Step 2: Set the denominator equal to zero

In order to find the values that must be excluded from the domain, set the denominator equal to zero and solve for x. This gives: \( x-3 = 0 \)

Step 3: Solve for x

Solving the equation \( x-3 = 0 \) for x, you find: \( x = 3 \)

Step 4: Write down the numbers that must be excluded from the domain

Thus, the number 3 must be excluded from the domain of this rational function, because it would make the denominator of the function equal to zero.

Key Concepts

Rational Function Domain

Understanding the domain of a rational function is vital for correctly working with these expressions. In simple terms, the domain refers to the set of all possible values that the independent variable can take on for the function to produce a real number.

For the rational function like the one in our exercise \( \frac{7}{x-3} \) , the domain includes all real numbers except for those that make the denominator equal to zero. Why? Because division by zero is undefined in mathematics; it is an operation without meaning in the real number system. Therefore, identifying values that lead to a zero in the denominator is crucial since these 'excluded values' help establish the domain of the function.

It's important to remember that while the numerator can be zero (since zero divided by any non-zero number is 0), it is solely the denominator that determines the domain's restrictions.

Excluded Values in Rational Expressions

Rational expressions, like the one we have in our problem \( \frac{7}{x-3} \) , consist of a numerator and a denominator, just like fractions. The excluded values are specific numbers that cannot be used for the variable because they would make the denominator zero.

To find these excluded values, simply set the denominator equal to zero and solve for the variable, as was done in the step-by-step solution: \( x-3 = 0 \) which gives \( x = 3 \) .

These excluded values are important for many reasons:

• They define the boundaries of the rational expression's domain.
• Knowing them helps avoid mathematical errors like division by zero.
• They are key to understanding the behavior of the graph of the rational function.

Always check for and exclude these values from the domain when working with rational expressions.

Solving Rational Equations

Solving rational equations, which contain one or more rational expressions, involves finding the value of the variable that makes the equation true. But, it’s not as straightforward as solving simple algebraic equations due to the presence of denominators that can't be zero.

The process usually includes these steps:

• Identify any excluded values by setting denominators to zero.
• Find a common denominator and multiply each term by it to eliminate fractions.
• Simplify the resulting equation and solve for the variable.
• Check your solutions to make sure they do not include the excluded values.

By carefully following these steps and being mindful of the excluded values, one can effectively solve rational equations and understand the possible solutions within their correct domain.