Problem 11 - College Algebra with Modeling and Visualization

Question

Determine whether \(f\) is a rational function and state its domain. $$ f(x)=4-\frac{3}{x+1} $$

Step-by-Step Solution

Answer

Yes, \(f(x)\) is a rational function. Domain: \(( -\infty, -1 ) \cup ( -1, \infty )\)."

Step 1: Identify Function Type

Examine the function given, \(f(x) = 4 - \frac{3}{x+1}\). A rational function is defined as a function that can be expressed as the quotient of two polynomials. In this case, the term \(\frac{3}{x+1}\) is a rational expression because it is the quotient of the constant polynomial 3 and the linear polynomial \(x+1\). Thus \(f(x)\) is a rational function as it can be rewritten in the form: \(f(x) = \frac{4(x+1) - 3}{x+1}\).

Step 2: Find Domain Restrictions

The domain of a rational function consists of all real numbers except those that make the denominator equal to zero. For the term \(\frac{3}{x+1}\), look for values of \(x\) that would make the denominator, \(x+1\), equal to zero. Setting \(x+1 = 0\) and solving for \(x\), we find \(x = -1\). Thus, the function \(f(x)\) is undefined at \(x = -1\).

Step 3: State the Domain

After identifying the restrictions, state the domain of the function. The domain of \(f(x) = 4 - \frac{3}{x+1}\) is all real numbers except \(x = -1\). In interval notation, the domain is expressed as \(( -\infty, -1 ) \cup ( -1, \infty )\).

Key Concepts

Domain of a Function

The domain of a function is essential because it tells us which values we can input into the function without causing any mathematical issues, like division by zero. In most functions, the domain includes all real numbers, but there are exceptions. Rational functions, which this article primarily discusses, usually have restrictions on their domains. Let's look at why. A rational function is a fraction where both the numerator and the denominator are polynomials. Since a rational function is defined as long as the denominator isn’t zero, our job is to find any values that make the denominator zero and exclude them from the domain. In the example, the rational part of the function is \( \frac{3}{x+1} \). Here, we set the denominator, \( x+1 \), equal to zero to find the restricted value:

• \( x + 1 = 0 \)
• So, \( x = -1 \)

This means the function is undefined at \( x = -1 \), and all other real numbers are part of the domain. The domain is then all real numbers except \( x = -1 \), expressed in interval notation as \( (-\infty, -1) \cup (-1, \infty) \). This notation highlights the idea of excluding \( x = -1 \) without any gaps in the allowed values of \( x \).

Function Types

Functions come in various forms, and understanding these can help with identifying how they behave and how to work with them. One common type is the polynomial function, which involves sums and products of variables raised to whole-number powers.However, in the case of rational functions, we deal with quotients of polynomials. A rational function looks like \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are both polynomials, and \( Q(x) eq 0 \). This definition is crucial as it outlines the core feature of rational functions: they must involve a division of polynomials. In our example, \( f(x) = 4 - \frac{3}{x+1} \), while it might not initially appear so, \( \frac{3}{x+1} \) is a rational function since it can be expressed as \( \frac{3}{1} \) divided by \( x+1 \). This format demonstrates the rational function's characteristic - containing a division by a polynomial. Recognizing function types, especially rational ones, is crucial for determining behavior like asymptotes and the overall shape of the graph, as well as finding domain restrictions.

Polynomials

Polynomials are the building blocks for many types of mathematical functions, including rational ones. A polynomial is made up of variables raised to different powers with their coefficients, where the exponents are non-negative integers. Polynomials can appear as simple expressions like \( 3x^2 + x - 5 \), or as constants like \( 3 \).When these expressions appear in rational functions, they’re usually in both the numerator and the denominator. This gives the function its distinct rational quality. In the example function, \( f(x) = 4 - \frac{3}{x+1} \), the polynomial in the denominator, \( x+1 \), plays a significant role in determining the function's domain. Because any polynomial can potentially become zero, it's important to analyze these expressions carefully.Polynomials allow us to perform operations like addition, subtraction, and multiplication within functions. However, the moment we introduce division by a polynomial, we transition into exploring rational functions - a concept with broader implications on the domain and behavior of a function on a graph. Understanding polynomials and their role in functions is vital, as these expressions are foundational in algebra and calculus. They are the tools through which mathematical relationships are expressed and analyzed.