Problem 14 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\frac{-3}{x+4}$$

Step-by-Step Solution

Answer

The domain of \( f(x) = \frac{-3}{x+4} \) is all real numbers except \( x = -4 \).

Step 1: Understand the Problem

We need to determine the domain of the function \( f(x) = \frac{-3}{x+4} \). The domain is the set of all possible values of \( x \) for which the function is defined.

Step 2: Identify Restrictions

Since the function is a rational function, it will be undefined for values that make the denominator equal to zero. Here, the denominator is \( x+4 \).

Step 3: Solve for Undefined Values

Set the denominator equal to zero and solve for \( x \):\[x+4 = 0\]\[x = -4\]

Step 4: Determine the Domain

Since \( x = -4 \) makes the denominator zero, \( x = -4 \) is not in the domain. Thus, the domain of the function is all real numbers except \( x = -4 \). In interval notation, this is:\(( -\infty, -4) \cup (-4, \infty)\).

Key Concepts

Rational Functions

A rational function is a type of function represented by the ratio of two polynomials. It's like a fractional expression where the numerator and denominator are both polynomials. Rational functions are important in many areas of mathematics as they can describe a wide variety of natural phenomena. However, they come with a key characteristic that makes them unique: they can be undefined for certain values of the variable.
When dealing with rational functions, it's crucial to look at the denominator because dividing by zero is not allowed in mathematics. This creates restrictions on the domain of the function. For example, in the function \( f(x) = \frac{-3}{x+4} \), the denominator \( x+4 \) cannot be zero. Identifying where the denominator equals zero helps us find the values that the function can't accept and thus helps in determining the domain.

Function Notation

Function notation may seem fancy, but it's actually quite straightforward. It is a way of representing functions in algebraic terms. It's written as \( f(x) \), where \( f \) is the name of the function, and \( x \) is the variable you input. Think of \( f(x) \) as the output you get when you plug a specific value of \( x \) into the function equation.
Function notation is incredibly useful because it clearly indicates which function is being used and what the input variable is. For example, in \( f(x) = \frac{-3}{x+4} \), we immediately know:

• "f" represents the function name.
• "x" is the variable whose values we manipulate.

This clarity helps prevent confusion, especially when working with multiple functions or when solving for specific outputs by plugging in various input values.

Interval Notation

Interval notation is a system of writing subsets of the real number line. It's a concise way of representing a set of numbers, including all the numbers between two endpoints. With interval notation, you can easily see which numbers are included and which numbers aren't. For example, the set of all real numbers except \( x = -4 \) is written as \( (-\infty, -4) \cup (-4, \infty) \).
When you see \(( -\infty, -4)\), it means the set includes all numbers less than \(-4\). The negative infinity symbol \((-\infty)\) indicates that there is no lower bound. Similarly, \((-4, \infty)\) includes all real numbers greater than \(-4\), and \((\infty)\) indicates no upper bound. The union symbol \((\cup)\) connects the two intervals, showing both are part of the domain.Here are some basic symbols used in interval notation:

• "(" or ")" indicates the number is not included in the interval.
• "[" or "]" indicates the number is included in the interval.