Problem 1 - College Algebra

Question

In Exercises \(1-8,\) find the domain of each rational function. $$f(x)=\frac{5 x}{x-4}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\frac{5 x}{x-4}\) is all real numbers except \(x = 4\).

Step 1: Identifying the Rational Function

The function given to us is \(f(x)=\frac{5 x}{x-4}\). This function is a rational function because it's in the form of a fraction where the numerator and the denominator are polynomials.

Step 2: Setting the Denominator Equal to Zero

The next step involves setting the denominator equal to zero to find the value that x cannot take. Therefore, you see that \(x-4 = 0\). Solving this we get \(x = 4\).

Step 3: Determining the Domain of the Function

The domain of a function includes all possible input values. The only value that x can't take is 4 because it would render the denominator equal to zero. Therefore, the domain of this function is all x (which belongs to real numbers) except x=4.

Key Concepts

Rational Functions

A rational function is essentially a fraction of two polynomials.

• The numerator and denominator of a rational function are both polynomials. For example, in the function \( f(x) = \frac{5x}{x-4} \), "5x" is the polynomial numerator, and "x-4" is the polynomial denominator.
• Rational functions are called such because they are expressed as the ratio of two polynomials. The word "rational" comes from "ratio" which signifies this relationship.
• Important characteristics of rational functions include their domains, asymptotes, and intercepts. Understanding these features helps in sketching the graph of rational functions.

Rational functions are commonly found in mathematical problems involving rates and proportions.

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients.

• A polynomial is formed by summing terms of the form \( ax^n \), where "a" is a coefficient and \( n \) is a non-negative integer exponent. For example, "5x" is a polynomial of degree 1.
• Polynomials are used as the building blocks in rational functions, serving as the numerators and denominators.
• The degree of a polynomial is determined by the highest power of the variable present in the expression. In \( 5x \), the degree is 1, as the highest power is 1.

Polynomials are foundational in algebra and calculus, often appearing in equations that describe curves and motions.

Function Domains

The domain of a function is the complete set of possible input values accepted by the function.

• In algebra, determining the domain requires considering any restrictions or limitations. In rational functions, the denominator cannot be zero, which creates exclusions in the domain.
• The domain is usually expressed in interval notation or set notation to indicate which values are included or excluded.
• For the function \( f(x) = \frac{5x}{x-4} \), the domain would exclude \( x = 4 \) since it would cause division by zero, leading to an undefined expression.

Understanding function domains is essential for correctly solving equations and determining where the function is valid.

Exclusions in Domain

Exclusions in the domain occur when certain values make the function undefined or problematic.

• The primary reason for exclusions in rational functions is a zero denominator, as division by zero is not possible.
• To find exclusions, set the denominator equal to zero and solve for the variable. This identifies the values that must be excluded from the domain.
• For \( f(x) = \frac{5x}{x-4} \), solving \( x-4 = 0 \) gives \( x = 4 \). This value is excluded from the domain, meaning the function is not defined at \( x = 4 \).

Identifying exclusions helps in understanding the behavior of the function and its graph, such as identifying asymptotes or discontinuities.