Problem 20 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\frac{7}{x^{2}-8 x-20}$$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is all real numbers except \( x = 10 \) and \( x = -2 \).

Step 1: Identify the Denominator

The function given is \( f(x) = \frac{7}{x^2 - 8x - 20} \). The domain excludes any \( x \) values that make the denominator zero. Start by identifying the denominator: \( x^2 - 8x - 20 \).

Step 2: Set Denominator Equal to Zero

To find the values of \( x \) that are not in the domain, set the denominator equal to zero: \( x^2 - 8x - 20 = 0 \).

Step 3: Factor the Quadratic Equation

Attempt to factor the quadratic equation: \( x^2 - 8x - 20 = 0 \). The equation factors into: \((x - 10)(x + 2) = 0\).

Step 4: Solve for Excluded Values

Using the factored form, solve each equation: \( x - 10 = 0 \) gives \( x = 10 \), and \( x + 2 = 0 \) gives \( x = -2 \). These are the values that make the denominator zero.

Step 5: Specify the Domain

Since the function is undefined at \( x = 10 \) and \( x = -2 \), the domain excludes these values. Therefore, the domain of \( f(x) \) is all real numbers except \( x = 10 \) and \( x = -2 \). This can be written as: \( x \in \mathbb{R}, x eq 10, x eq -2 \).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra. They are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The highest exponent in the equation is 2, which results in the name quadratic, from the Latin "quadratus," meaning square. Quadratic equations can appear in various scenarios, such as physics and engineering problems, where they model parabolic paths.

To solve quadratic equations, we often look for roots or solutions where the function equals zero. Finding these roots can be done through factoring, using the quadratic formula, or completing the square. Each method has its own use-case and depending on the form of the given quadratic equation, one may be more advantageous than the others.

When we set a quadratic equation equal to zero, as we did with the function's denominator in the exercise, we identify those \( x \) values that cause the equation to equal zero. Solving these helps us understand more about the exclusion of certain values in a function's domain. These excluded values, derived from the equation equaling zero, highlight where the function may be undefined, such as division by zero in rational functions.

Factoring Polynomials

Factoring polynomials is a key skill in engineering and mathematics. It involves rewriting a polynomial as a product of its simpler polynomial factors. In essence, it's about breaking down complex expressions into pieces that multiply together to form the original polynomial.

Consider the example from the exercise: \( x^2 - 8x - 20 = 0 \). Here, the quadratic polynomial is factored into \((x - 10)(x + 2)\). Why is factoring useful?

• It simplifies solving quadratic equations by converting them into linear factors.
• Once in factored form, solving for \( x \) becomes straightforward by setting each factor equal to zero individually.
• Factoring aids in graphing polynomial functions, providing insight into their roots, which are where the graph intersects the x-axis.

To factor a simple quadratic, find two numbers whose product equals \( c \) (the constant term) and whose sum equals \( b \) (the linear coefficient).
In our example, \( -10 \) and \( 2 \) multiply to \( -20 \) and sum to \( -8 \), making them the key to factoring. Mastering this skill is essential for tackling more complex equations effectively.

Excluded Values in Domain

The domain of a function includes all possible input values (x-values) for which the function is defined. For rational functions, like \(f(x) = \frac{7}{x^2 - 8x - 20}\), we need to be mindful of points where the function may not be defined, such as where the denominator equals zero.

These specific x-values that make the denominator zero create excluded values in the domain. Let's see why: If \( x = 10 \) or \( x = -2 \), the denominator becomes zero, leading to division by zero, which is undefined in mathematics. Therefore, these values aren't included in the domain.

Defining a function's domain might include:

• Identifying any restrictions based on mathematical operations (e.g., division by zero).
• Evaluating the set of real numbers or a specific interval that excludes troublesome points.

In the provided exercise, solving \( (x - 10)(x + 2) \) reveals \( x = 10 \) and \( x = -2 \) as excluded values. So, the domain is all real numbers except these points, which ensures that the function remains well-defined and usable in calculations.