Problem 6 - College Algebra

Question

In Exercises \(1-8,\) find the domain of each rational function. $$h(x)=\frac{x+8}{x^{2}-64}$$

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\frac{x+8}{x^{2}-64}\) is all real numbers except -8 and 8.

Step 1: Identify the Denominator

The denominator of the function is \(x^{2}-64\).

Step 2: Find when the Denominator Equals Zero

To find the values of x that we must exclude from the domain, we set the denominator equal to zero and solve for x. \(x^{2} -64 = 0\), this is a difference of squares which can be expressed as \((x-8)(x+8) = 0\). Therefore, \(x = -8\) or \(x = 8\).

Step 3: Express the Domain

The domain of the function is all real numbers except for the numbers that make the denominator zero. These numbers are \(x = -8\) and \(x = 8\). Thus, the domain of \(h(x)\) is all real numbers except -8 and 8.

Key Concepts

Rational Functions

Rational functions are a type of function expressed as the quotient of two polynomials. This can be visualized as a fraction where the numerator and the denominator are both polynomial expressions. A typical rational function looks like this:

• Numerator: a polynomial, e.g. \( x + 8 \)
• Denominator: another polynomial, e.g. \( x^2 - 64 \)

The function \( h(x) = \frac{x+8}{x^2-64} \) is an excellent example of a rational function. The numerator is \( x + 8 \) and the denominator is \( x^2 - 64 \). When working with rational functions, one critical task is to determine the values that make the denominator equal to zero. This is essential as it directly affects the domain, or the set of permissible input values for the function.

Denominator Zero

A fundamental characteristic of rational functions is the restriction on values where the denominator equals zero. In mathematics, division by zero is undefined, which means any x-values that turn the denominator into zero must be excluded from the domain.To find these values, you set the denominator equal to zero and solve for \( x \).

• The equation \( x^2 - 64 = 0 \) determines these x-values.
• Simplifying the equation helps identify the critical points: \( (x - 8)(x + 8) = 0 \).

Solving for \( x \) gives \( x = 8 \) and \( x = -8 \), meaning these values are not permitted in the domain of the function. Remember, identifying where the denominator equals zero is crucial when working with rational functions.

Difference of Squares

The concept of difference of squares is a special type of polynomial factoring which is very useful when working with rational functions. For instance, the expression \( x^2 - 64 \) can be factored because it is a difference of squares. This means it takes the form:\[ a^2 - b^2 = (a - b)(a + b) \]In the equation \( x^2 - 64 \), consider it as \( x^2 - 8^2 \), enabling factoring into \((x-8)(x+8)\).

• This factoring helps identify the points where the denominator becomes zero by setting each factor equal to zero.
• Solving these gives \( x = 8 \) and \( x = -8 \), which need to be excluded from the domain.

Understanding and using difference of squares provides a streamlined approach to solving such rational functions effectively.

Real Numbers Exclusion

When we discuss the domain of a rational function, we typically mean the range of real numbers allowed as inputs. However, as we've seen, certain values must be excluded due to division by zero.Once determined that \( x = 8 \) and \( x = -8 \) make the denominator zero, these values are excluded from the domain. Thus, the domain of \( h(x) = \frac{x+8}{x^2-64} \) includes all real numbers except these two.

• To express the domain in interval notation, note that the function is defined from: \(( -\infty, -8 ) \cup (-8, 8) \cup (8, \infty)\).

Excluding certain real numbers ensures that the function remains valid and defined across the entire domain. Domain analysis often involves finding these exclusions to maintain the mathematical integrity of the function.