Problem 31 - Elementary and Intermediate Algebra

Question

Find the domain of each rational function. Express your answer in words and using interval notation. See Example 2. $$f(x)=\frac{x^{2}+3 x+2}{x^{2}-x-56}$$

Step-by-Step Solution

Answer

All real numbers except -7 and 8; domain in interval notation: \( (-\infty, -7) \cup (-7, 8) \cup (8, \infty) \).

Step 1: Identify the Denominator

First, determine the denominator of the function which is \( x^2 - x - 56 \). The domain excludes values that make the denominator zero, so we need to find when \( x^2 - x - 56 = 0 \).

Step 2: Factor the Quadratic

To find the values that make the denominator zero, factor the quadratic expression. Start by finding two numbers that multiply to \(-56\) and add up to \(-1\). The numbers \(7\) and \(-8\) work, so the denominator factors into \((x - 8)(x + 7)\).

Step 3: Solve the Factored Equation

Set each factor equal to zero to find the values that make the denominator zero: \(x - 8 = 0\) and \(x + 7 = 0\). Solve these equations to find \(x = 8\) and \(x = -7\).

Step 4: Determine the Domain

Since the function is undefined when the denominator is zero, we must exclude \(x = 8\) and \(x = -7\) from the domain. In interval notation, the domain is \( (-\infty, -7) \cup (-7, 8) \cup (8, \infty) \). In words, the domain consists of all real numbers except \(-7\) and \(8\).

Key Concepts

Domain of a Function

Every function has a range of inputs called its "domain." For many elementary functions, the domain is simply all real numbers. However, for rational functions (functions expressed as the division of two polynomials), we need to be more careful. The domain of a rational function excludes any input values that make the denominator of the function equal to zero since division by zero is undefined.

• To find the domain, you start by identifying the denominator of the rational function.
• Then, find the values for which this denominator is zero. These values are excluded from the domain.

In our example, the denominator is \(x^2 - x - 56\). After determining the values \(x = 8\) and \(x = -7\) make this expression zero, these are excluded from the domain of the function. Thus, the domain excludes \(x = 8\) and \(x = -7\).

Interval Notation

Interval notation is a method of describing a range of values in a concise form. It's often used to express domains and ranges of functions.
In interval notation:

• Parentheses \(()\) are used to indicate values that are not included in the set (open intervals).
• Brackets \([]\) are used to indicate values that are included in the set (closed intervals).

To express the domain of a function using interval notation, outline the segments of real numbers that are included, using union symbols \(\cup\) to link multiple intervals if needed.
For the function \(f(x) = \frac{x^{2}+3 x+2}{x^{2}-x-56}\), we exclude \(x = 8\) and \(x = -7\), so the domain in interval notation is \((-\infty, -7) \cup (-7, 8) \cup (8, \infty)\). This reads that the domain includes all numbers from negative infinity to \(-7\), from \(-7\) to \(8\), and from \(8\) to infinity, skipping \(-7\) and \(8\) themselves.

Factoring Quadratics

Factoring quadratics is a method of rewriting a quadratic expression as a product of linear factors. This is often needed when dealing with rational functions to find values that make the denominator zero.

• The general form of a quadratic is \(ax^2 + bx + c\).
• To factor, you need to find two numbers that multiply to \(c\) and add to \(b\).

In our exercise, the quadratic \(x^2 - x - 56\) is factored by looking for two numbers that multiply to \(-56\) (the constant term) and add to \(-1\) (the linear coefficient). These numbers are \(7\) and \(-8\). Therefore, the quadratic factors into \((x - 8)(x + 7)\).
Factoring is essential as it helps identify the values of \(x\) the denominator can take that would render the whole function undefined.