Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of the rational functions. $$ f(x)=\frac{x^{2}+4 x-3}{x^{4}-5 x^{2}+4} $$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is all real numbers except \( x = -2, -1, 1, 2 \).

Step 1: Identify the Rational Function

Examine the given rational function. The function is \( f(x) = \frac{x^2 + 4x - 3}{x^4 - 5x^2 + 4} \). The domain of a rational function includes all real numbers except where the denominator is zero.

Step 2: Set Denominator Equal to Zero

Set the denominator equal to zero to find the values of \( x \) for which the function is not defined. Solve \( x^4 - 5x^2 + 4 = 0 \).

Step 3: Substitute and Simplify

Let \( y = x^2 \), then the equation becomes \( y^2 - 5y + 4 = 0 \). This is a quadratic equation in terms of \( y \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve the equation \( y^2 - 5y + 4 = 0 \), where \( a = 1 \), \( b = -5 \), and \( c = 4 \).

Step 5: Calculate the Solutions for 'y'

Calculate the solutions for \( y \) using the quadratic formula: \( y = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm 3}{2} \). This gives the solutions \( y = 4 \) and \( y = 1 \).

Step 6: Translate 'y' Back to 'x'

Recall that \( y = x^2 \). So, we have \( x^2 = 4 \) and \( x^2 = 1 \). Solve these equations to find \( x = \pm 2 \) and \( x = \pm 1 \).

Step 7: Define the Domain

The domain of \( f(x) \) consists of all real numbers except where the function is undefined. Thus, exclude \( x = \pm 2, \pm 1 \) from the domain. Therefore, the domain of \( f(x) \) is all real numbers except \( x = -2, -1, 1, 2 \).

Key Concepts

Domain of Functions

When working with rational functions, it's crucial to first identify the domain of the function. The domain refers to all possible values of the variable for which the function is defined. Rational functions are expressed as the division of two functions, typically polynomials. The restriction on the domain arises from the need to avoid division by zero. Any values of the variable that make the denominator zero are excluded from the domain.

To find these restrictions, we set the denominator equal to zero and solve the resulting equation. For example, in the rational function given by \( f(x) = \frac{x^2 + 4x - 3}{x^4 - 5x^2 + 4} \), we would set \( x^4 - 5x^2 + 4 = 0 \). Solving this equation reveals the values that must be excluded from the domain. Understanding the domain is essential as it tells us the range of values that we can safely plug into the function without causing undefined expressions.

Quadratic Equation

An important step in solving equations, especially those related to rational functions, is understanding and solving quadratic equations. Quadratic equations are polynomial equations of the second degree, typically written in the form \( ax^2 + bx + c = 0 \).

To solve a quadratic equation, one common approach is to use the quadratic formula:

• \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula allows us to find the roots or solutions of the quadratic equation, where \( a \), \( b \), and \( c \) are constants from the standard form of the equation. In our exercise, we transformed the expression \( x^4 - 5x^2 + 4 \) into a quadratic equation by setting \( y = x^2 \), leading to \( y^2 - 5y + 4 = 0 \). With \( a = 1 \), \( b = -5 \), and \( c = 4 \), substituting these into the formula gives us the values for \( y \). Understanding how to derive and solve quadratic equations is fundamental to interpreting and solving rational functions.

Solving Equations

Solving equations in mathematics involves finding the value(s) of the variable that satisfy the equation. When dealing with polynomial or quadratic equations, several methods are available, such as factoring, completing the square, or using the quadratic formula.

For our particular problem, we used the substitution method by letting \( y = x^2 \) to simplify the quartic equation into a quadratic one. Once in quadratic form, the quadratic formula was applied to find the solutions for \( y \). We found that \( y = 4 \) and \( y = 1 \). Translating these back to terms of \( x \) involved solving the equations \( x^2 = 4 \) and \( x^2 = 1 \), giving solutions \( x = \pm 2 \) and \( x = \pm 1 \).

The art of solving equations entails not only carrying out the mathematical manipulations but also interpreting these results correctly in the context of the problem, such as excluding certain values from the domain of a rational function.

Real Numbers Exclusion

In the context of solving rational functions, understanding real numbers exclusion is pivotal. This concept involves identifying real numbers that are not part of the domain of a function.

For a rational function, these are typically the real numbers that make the denominator zero. By solving the denominator equation \( x^4 - 5x^2 + 4 = 0 \), we identified the critical points: \( x = -2, -1, 1, 2 \). These values, when substituted back into the denominator, yield zero, making the function undefined at these points.

Thus, the process of real numbers exclusion ensures our function is defined and avoids mathematical errors due to division by zero. Ensuring these exclusions are made correctly prevents undefined behavior and enhances accuracy in mathematical computations involving rational functions.