Problem 19 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\frac{14}{x^{2}+3 x-40}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, -8) \cup (-8, 5) \cup (5, \infty)\)

Step 1: Understand Rational Functions

The function given is \(f(x) = \frac{14}{x^2 + 3x - 40}\). This is a rational function because it is a fraction with a polynomial in the numerator and denominator. To find the domain of a rational function, identify values of \(x\) that would make the denominator zero.

Step 2: Set Denominator to Zero

To find these problematic values, set the denominator equal to zero: \(x^2 + 3x - 40 = 0\). Solving this equation will give us values of \(x\) that are not included in the domain.

Step 3: Solve the Quadratic Equation

Solve the quadratic equation \(x^2 + 3x - 40 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 3\), and \(c = -40\).

Step 4: Calculate Discriminant and Solve

Calculate the discriminant: \(b^2 - 4ac = 3^2 - 4(1)(-40) = 9 + 160 = 169\). Then apply the quadratic formula: \[ x = \frac{-3 \pm \sqrt{169}}{2 \times 1} \] This simplifies to \[ x = \frac{-3 \pm 13}{2} \].

Step 5: Find Exact Zeroes

Continuing from the quadratic formula solution, we have:- \( x = \frac{-3 + 13}{2} = 5 \)- \( x = \frac{-3 - 13}{2} = -8 \)Thus, the values that make the denominator zero are \(x = 5\) and \(x = -8\).

Step 6: Determine the Domain

The domain of the function \(f(x)\) includes all real numbers except those that make the denominator zero. Therefore, the domain is all real numbers \(x\) such that \(x eq 5\) and \(x eq -8\). In interval notation, this is expressed as \((-\infty, -8) \cup (-8, 5) \cup (5, \infty)\).

Key Concepts

Domain of a Function

When we talk about the domain of a function, we are referring to all the input values (typically represented by \(x\)) for which the function is defined. In simpler terms, it's the collection of all possible \(x\) values you can plug into the function without causing any mathematical issues, such as division by zero or taking the square root of a negative number in the real number system.

For rational functions, which involve fractions with polynomials in the numerator and denominator, you want to pay attention to the denominator. If the denominator equals zero, the function becomes undefined. To find the domain of a rational function:

• Set the denominator equal to zero.
• Solve the resulting equation to find the \(x\) values that make the denominator zero.
• Exclude these \(x\) values from the domain.

By doing so, you identify all the \(x\) values that the function cannot take, ensuring the function is defined for all values in its domain.

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. Its standard form is given by \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Solving a quadratic equation will often involve finding the roots, or the \(x\) values that make the equation equal to zero.

The quadratic formula is a powerful tool that helps us find these roots:

• Identify the coefficients \(a\), \(b\), and \(c\).
• Calculate the discriminant \(b^2 - 4ac\). The value of the discriminant can tell us the nature of the roots:
  • If it is positive, there are two distinct real roots.
  • If it is zero, there is one real root (the roots are repeated).
  • If it is negative, the roots are complex and not real.
• Apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Understanding how to solve quadratic equations is crucial because they frequently appear in various math problems, including determining the domain of rational functions.

Interval Notation

Interval notation is a way of representing a set of numbers as an interval on the number line. It's often used to specify the domain of functions because it clearly shows which numbers are included or excluded from the domain.

Here are some basic principles of interval notation:

• Parentheses \(()\) indicate that an endpoint is not included in the interval.
• Brackets \([]\) indicate that an endpoint is included in the interval.
• The symbol \(\infty\) (infinity) or \(-\infty\) (negative infinity) is used to represent intervals that extend indefinitely in either direction, always with parentheses since infinity is not a number that can be reached.
• A union symbol \(\cup\) is used to combine two or more intervals.

For example, the domain \((-\infty, -8) \cup (-8, 5) \cup (5, \infty)\) indicates that the function is defined for all real numbers except \(x = -8\) and \(x = 5\), which are not included in the domain. Interval notation provides a concise and efficient way to describe complex sets of numbers.