Problem 24 - Intermediate Algebra

Question

Find the domain of each rational function. Express your answer in words and using interval notation. $$ f(x)=\frac{x^{2}+36}{x^{2}-36} $$

Step-by-Step Solution

Answer

Domain is all real numbers except 6 and -6. In interval notation: \((-\infty, -6) \cup (-6, 6) \cup (6, \infty)\).

Step 1: Understand the Rational Function

The given function is a rational function, which is in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. For \( f(x) = \frac{x^2 + 36}{x^2 - 36} \), the numerator \( P(x) = x^2 + 36 \), and the denominator \( Q(x) = x^2 - 36 \). The domain of a rational function includes all real numbers except those for which the denominator equals zero.

Step 2: Set the Denominator Equal to Zero

To find the values that are not in the domain, set the denominator equal to zero: \( x^2 - 36 = 0 \). Solve this equation to determine the restrictions on \( x \).

Step 3: Solve the Equation for x

Factor the equation \( x^2 - 36 = 0 \) as a difference of squares: \( (x - 6)(x + 6) = 0 \). This gives the solutions \( x - 6 = 0 \) and \( x + 6 = 0 \). Therefore, \( x = 6 \) and \( x = -6 \).

Step 4: Determine the Domain

Since \( x = 6 \) and \( x = -6 \) make the denominator zero, these values are not in the domain. The domain includes all real numbers except \( 6 \) and \( -6 \).

Step 5: Express the Domain in Words and Interval Notation

In words, the domain of \( f(x) \) is all real numbers except 6 and -6. In interval notation, the domain is \((-\infty, -6) \cup (-6, 6) \cup (6, \infty)\).

Key Concepts

Rational Functions

A rational function is a type of function that is expressed as the quotient of two polynomials. It takes the form \( f(x) = \frac{P(x)}{Q(x)} \). Here, \( P(x) \) and \( Q(x) \) are polynomials. Polynomials consist of terms which each have a constant multiplied by a variable raised to a non-negative integer exponent. Rational functions are defined for all real numbers except where the denominator is zero. This is because division by zero is undefined in mathematics, leading to a restriction in the domain of the function. For example, consider the rational function \( f(x) = \frac{x^2 + 36}{x^2 - 36} \). The numerator \( x^2 + 36 \) does not affect the domain, but the denominator \( x^2 - 36 \) needs to be excluded from equal to zero. Thus, solving \( x^2 - 36 = 0 \) helps us derive these restrictions in our domain. This ensures the function operates under valid mathematical conditions.

Interval Notation

Interval notation is a mathematical notation used to describe a set of numbers, specifically a range of numbers between two endpoints. It's efficient and concise for expressing domains of functions. Interval notation uses parentheses \(()\) and brackets \([]\) to indicate whether endpoints are included or excluded.

• Parentheses \(()\) are used when endpoints are not included.
• Brackets \([]\) are used when endpoints are included.

Consider the domain of the rational function \( f(x) = \frac{x^2 + 36}{x^2 - 36} \), which is all real numbers except \( x = 6 \) and \( x = -6 \). In interval notation, we write the domain as \((-infty, -6) \cup (-6, 6) \cup (6, infty)\). This expression indicates all numbers from negative infinity to -6, excluding -6, combined with numbers from -6 to 6, excluding both, and continuing from 6 to infinity, excluding 6.

Polynomials

Polynomials form the basis of rational functions, serving as both numerators and denominators. They are expressions comprising variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial looks like \( P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( a_n, a_{n-1}, ..., a_1, a_0 \) are constants called coefficients, and \( n \) is a non-negative integer.In the rational function \( f(x) = \frac{x^2 + 36}{x^2 - 36} \), both \( x^2 + 36 \) and \( x^2 - 36 \) are polynomials. The complexity of solving for the domain in rational functions arises mainly from dealing with their polynomial denominators. Specifically, identifying zero in the denominator's polynomial helps dictate the range of permissible values in rational functions.