Problem 9 - College Algebra

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 the function is all real numbers except \(x = -2, -1, 1, 2\).

Step 1: Understand the function and its components

The function given is a rational function, which means it is the ratio of two polynomials. In the function \(f(x) = \frac{x^2 + 4x - 3}{x^4 - 5x^2 + 4}\), the numerator is \(x^2 + 4x - 3\) and the denominator is \(x^4 - 5x^2 + 4\).

Step 2: Determine the restriction for the denominator

For any rational function, the denominator cannot be zero as this would make the function undefined. Therefore, to find the domain, we need to solve for \(x\) such that the denominator \(x^4 - 5x^2 + 4 = 0\).

Step 3: Solve the equation for the denominator

To solve \(x^4 - 5x^2 + 4 = 0\), we can substitute \(u = x^2\), transforming the equation into a quadratic form \(u^2 - 5u + 4 = 0\).

Step 4: Find the roots of the quadratic equation

Solve the quadratic equation \(u^2 - 5u + 4 = 0\) using the quadratic formula: \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 1\), \(b = -5\), and \(c = 4\). Calculate the discriminant: \[ b^2 - 4ac = (-5)^2 - 4(1)(4) = 25 - 16 = 9 \]Thus, the solutions for \(u\) are:\[ u = \frac{5 \pm \sqrt{9}}{2} \]\[ u = \frac{5 \pm 3}{2} \]\[ u_1 = 4, u_2 = 1 \]

Step 5: Substitute back to find x-values

Since \(u = x^2\), we substitute back to find the \(x\) values:- When \(u = 4\), \(x^2 = 4\), hence \(x = \pm 2\).- When \(u = 1\), \(x^2 = 1\), hence \(x = \pm 1\).

Step 6: Determine the domain of the function

The values \(x = \pm 2\) and \(x = \pm 1\) make the denominator zero, thus they are not in the domain of \(f(x)\). Therefore, the domain consists of all real numbers except \(x = -2, -1, 1, 2\). The domain in interval notation is:\( (-\infty, -2) \cup (-2, -1) \cup (-1, 1) \cup (1, 2) \cup (2, \infty) \)

Key Concepts

Polynomials

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, these are expressions composed of variables and coefficients, specifically raised to whole number exponents. For example, expressions like \(x^2 + 4x - 3\) and \(x^4 - 5x^2 + 4\) are polynomials because they involve sums of terms with variables raised to the power of 2 and 4, respectively. Each term in a polynomial is known as a monomial. Polynomials are classified based on their degree, where the degree is the highest power of the variable in the expression. Understanding polynomials is essential in finding the domain of rational functions since rational functions are defined as the ratio of two polynomials. Polynomials are continuous and defined everywhere on the real number line, which offers flexibility in various mathematical operations.
Key characteristics of polynomials include:

• Their coefficients can be any real number.
• Each term is a product of a constant and a non-negative integer power of a variable.
• They are defined and continuous everywhere on the real line unless in the denominator of a function, like in rational functions.

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. It takes the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In solving rational function domains, the denominator needs to be tested against zero, which often involves simplifying it into a quadratic equation format.
For example, in the original solution, the denominator \(x^4 - 5x^2 + 4\) is transformed into a quadratic in terms of \(u\), \(u^2 - 5u + 4 = 0\), where \(u = x^2\). Quadratic equations are commonly solved using the quadratic formula: \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The term under the square root, \( b^2 - 4ac \), is known as the discriminant. The discriminant helps in determining the nature and number of the roots of the quadratic equation:

• If it is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root.
• If it is negative, the equation has no real roots, only complex.

Interval Notation

Interval notation is a concise way of expressing a set of numbers or intervals. It is widely used in mathematics to define the domain (the set of all possible inputs) of functions, and it gives us a clearer picture of number ranges excluded or included in a particular function. Interval notation uses brackets \([\,]\) and parentheses \((\,)\) to denote closed and open intervals.
For example, the domain determined in the solution uses interval notation to exclude certain x-values that make the denominator zero: \( (-\infty, -2) \cup (-2, -1) \cup (-1, 1) \cup (1, 2) \cup (2, \infty)\) This expression tells us that all real numbers except for \(-2, -1, 1,\) and \(2\) are in the domain. As a general rule of thumb:

• Parentheses \( ( \) or \( ) \) indicate that an endpoint is not included in the interval (open interval).
• Brackets \( [ \) or \( ] \) indicate that an endpoint is included (closed interval).

You cannot include points where the function is undefined — these points are always excluded using parenthesis.

Function Undefined

A function becomes undefined when its mathematical expression does not yield a finite, real number result. For rational functions, this typically happens when the denominator equals zero, as division by zero is mathematically undefined.
In our exercise, we deal with a rational function \(f(x)=\frac{x^2+4x-3}{x^4-5x^2+4}\). This particular function is undefined for any \(x\) values that make the denominator zero. Through solving \(x^4 - 5x^2 + 4 = 0\), and substituting to return to \(x\), we identify the precise \(x = \pm 2\) and \(x = \pm 1\) as points where the rational function is undefined. Recognizing where a function becomes undefined is crucial in assessing its domain, as it helps us understand where the function "breaks" on the real number line, providing insights into behavioral limits and possible asymptotes of graphs.
Key points to consider include:

• Always solve the denominator equal to zero to find where the function is undefined.
• Exclude these values from the domain to prevent division errors.

Understanding where a function is undefined will help you accurately describe its domain and any asymptotic behavior.