Problem 4 - College Algebra

Question

Find all numbers that must be excluded from the domain of each rational expression. $$ \frac{x+7}{x^{2}-49} $$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the rational function are x = 7 and x = -7.

Step 1: Identify the Denominator

The denominator of the given function is \(x^{2}-49\). Any number which makes the denominator equal to zero is excluded from the domain because division by zero is undefined.

Step 2: Solve for Zero

Set the denominator equal to zero, and solve for x: \(x^{2}-49= 0\). This can be factored to \((x-7)(x+7) = 0\). Setting these factors equal to zero, we find x = 7 and x = -7. These are roots of the polynomial.

Step 3: Identify Excluded Values

The values x = 7 and x = -7 are excluded from the domain as they would result in the function \(\frac{x+7}{x^{2}-49}\) being undefined.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (usually "x" values) that the function can accept without causing any undefined behavior. For a rational expression like \( \frac{x+7}{x^{2}-49} \), the domain is limited by values that make the denominator zero. This is because division by zero is undefined in mathematics.

To determine the domain, you need to identify the values of \(x\) that make the denominator equal to zero. From the original exercise, the denominator is \(x^2 - 49\). By setting \(x^2 - 49 = 0\), you can solve for the values of \(x\) that are not allowed in the domain. These values make the expression undefined and should be excluded.

• The domain excludes \(x = 7\) and \(x = -7\).
• All other real numbers are included in the domain.

This method ensures we only use values that keep the function behaving properly without errors.

Undefined Math Expressions

Undefined math expressions occur typically when there is a division by zero. In rational expressions, this is a key concern as any value making the denominator zero causes the entire expression to be undefined. In our exercise \(\frac{x+7}{x^2 - 49}\), setting \(x^2 - 49 = 0\) helps identify where undefined expressions occur.

After factoring \(x^2 - 49\) into \((x-7)(x+7)\), we see that setting either factor to zero reveals the values \(x = 7\) and \(x = -7\).
These are the exact points where:

• \(x = 7\) results in \(7^2 - 49 = 0\)
• \(x = -7\) results in \((-7)^2 - 49 = 0\)

Under these conditions, the denominator becomes zero, making the expression undefined and therefore unsuitable for inclusion in the domain.

Solving Quadratic Equations

Solving quadratic equations is a useful skill when working with rational expressions. A typical quadratic equation takes the form of \(ax^2 + bx + c = 0\). In the exercise, the equation \(x^2 - 49 = 0\) represents a specific type of quadratic called a "difference of squares."

The general formula works by breaking down the equation into simpler factors. This can be done for equations like \(x^2 - 49\) by recognizing it as \((x - 7)(x + 7) = 0\). Each factor is then set to zero:

• \(x - 7 = 0\) yields \(x = 7\)
• \(x + 7 = 0\) yields \(x = -7\)

By solving these, we find both "roots" of the equation. Understanding how to solve quadratics can help tremendously when finding the domain of rational expressions, as it highlights where potential problems like division by zero might occur.