Problem 17 - Elementary Algebra

Question

For the following problems, find the domain of each of the rational expressions. $$ \frac{-x+4}{x^{2}-36} $$

Step-by-Step Solution

Answer

Answer: The domain of the given rational expression is $$(-\infty, -6) \cup (-6, 6) \cup (6, \infty)$$, which means all real numbers except for -6 and 6.

Step 1: Identify the denominator

The denominator of the given rational expression is: $$ x^2 - 36 $$

Step 2: Determine when the denominator equals to zero

We need to find values of \(x\) that make the denominator equal to zero, because these values won't be included in the domain. $$ x^2 - 36 = 0 $$ Notice that this is a difference of squares which can be factored as: $$ (x-6)(x+6) = 0 $$ Now, we solve for \(x\): $$ x-6 = 0 \Rightarrow x = 6 \\ x+6 = 0 \Rightarrow x = -6 $$ So, the values that make the denominator zero are \(x = -6\) and \(x = 6\).

Step 3: Determine the domain

Since the values of \(x = -6\) and \(x = 6\) make the denominator zero and division by zero is undefined, we must exclude them from the domain of the rational expression. The domain of the rational expression is the set of all real numbers excluding -6 and 6: $$ \text{Domain} = \{x \in \mathbb{R} \mid x \neq -6, x \neq 6\} $$ The domain can also be written in interval notation as: $$ \text{Domain} = (-\infty, -6) \cup (-6, 6) \cup (6, \infty) $$

Key Concepts

Denominator of a Rational Expression

Understanding the denominator of a rational expression is crucial when calculating its domain. A rational expression is a fraction wherein both the numerator and the denominator are polynomials. The key point to remember is that a denominator can never be zero since division by zero is undefined and would make the entire expression undefined. Hence, for the expression \( \frac{-x+4}{x^{2}-36} \), we focus on the denominator \( x^{2}-36 \) and find the values of \( x \) that would make it equal to zero. These specific values are then excluded from the domain of the expression.

While solving such problems, we perform a zero product property check. This involves equating the denominator to zero and solving for variable \( x \). In the given step-by-step solution, the identification and analysis of the denominator are the preliminary and most crucial steps to finding the domain of the rational expression.

Difference of Squares

A difference of squares is a special algebraic form that students frequently encounter in mathematics, especially when working with rational expressions and their domains. It is an expression of the form \( a^{2} - b^{2} \), which can be factored into \( (a+b)(a-b) \).

In our example, the denominator \( x^{2} - 36 \) is a difference of squares because it can be rewritten as \( (x)^{2} - (6)^{2} \) which factors into \( (x + 6)(x - 6) \). The zero product property tells us if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero to determine the critical values that are excluded from the domain: \( x = 6 \) and \( x = -6 \). Recognizing a difference of squares allows us to factor quickly and find these critical values efficiently.

Interval Notation

Interval notation is a system of writing subsets of the number line. It is particularly useful in describing domains of functions, including rational expressions. When we say that the domain of the given rational expression \( \frac{-x+4}{x^{2}-36} \) is \( (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \), we're using interval notation to describe all real numbers except -6 and 6.

The parentheses, '(', and ')', indicate that the endpoints are not included in the interval, signifying that \( x \) cannot equal -6 or 6, as these make the denominator zero. The union, '\( \cup \)', signifies that the domain is everything in the first interval, everything in the second interval, and everything in the third interval combined. The 'infinity' symbols are used to show that there is no upper or lower bound in those directions. If the domain did include the endpoints, we would use brackets, '[', and ']'. This clear and concise way of expressing ranges is very useful in mathematics and helps avoid any ambiguities in understanding the domain of rational expressions.