Problem 17 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$h(x)=\frac{2}{(x+1)(x-4)}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, -1) \cup (-1, 4) \cup (4, \infty)\).

Step 1: Understanding the Function

The given function is a rational function expressed as \( h(x) = \frac{2}{(x+1)(x-4)} \). A rational function is in the form of \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

Step 2: Identify the Denominator

The denominator of the function \( h(x) = \frac{2}{(x+1)(x-4)} \) is \( (x+1)(x-4) \). This component is crucial for determining the domain, as any value of \( x \) that makes the denominator zero will be excluded from the domain.

Step 3: Set the Denominator to Zero

To find the values that make the denominator zero, set \( (x+1)(x-4) = 0 \).

Step 4: Solve for x

Solve the equation \( (x+1)(x-4) = 0 \) to find the x-values that make the denominator zero. We have two factors to consider separately: \( x+1=0 \) and \( x-4=0 \).

Step 5: Solve Each Factor

1. Solve \( x+1=0 \): \( x = -1 \).2. Solve \( x-4=0 \): \( x = 4 \).

Step 6: State the Domain

The domain of the function \( h(x) \) includes all real numbers except the values that make the denominator zero. Thus, the domain is all real numbers except \( x = -1 \) and \( x = 4 \). In interval notation, this is: \( (-\infty, -1) \cup (-1, 4) \cup (4, \infty) \).

Key Concepts

Understanding Rational Functions

Rational functions are special types of mathematical functions formed by the ratio of two polynomials. Imagine a rational function as something like a fraction, where there is a top part (called the numerator) and a bottom part (called the denominator). For the function \( h(x) = \frac{2}{(x+1)(x-4)} \),

• the numerator is simply the number 2,
• and the denominator is the expression \((x+1)(x-4)\).

Rational functions are fascinating because they involve relationships between variables, and we can learn a lot about these relationships by exploring their domains and graphs.
Rational functions have some interesting properties, and one of the most critical things to understand is that they are defined only when the denominator is not zero. This is because division by zero is undefined in mathematics.

Interval Notation for Domains

Interval notation is a compact way of describing sets of numbers which make up the domain of a function. When talking about the domain of a function, especially rational ones, interval notation helps in conveying which numbers are included as possible inputs for the function.
The symbol \( (-\infty, 4) \) means all numbers less than 4, and \( (4, \infty) \) means all numbers greater than 4. When we have breaks in the sets or numbers skipped due to exclusions (like preventing division by zero in rational functions), we use the union symbol \( \cup \) to combine them.
For our function \( h(x) = \frac{2}{(x+1)(x-4)} \), the domain in interval notation is \((-\infty, -1) \cup (-1, 4) \cup (4, \infty)\). This indicates that every real number is allowed except \( -1 \) and \( 4 \), fitting them neatly into this unified format.

Exclusion of Real Numbers in Domains

In the world of functions, especially rational ones, certain real numbers are excluded from the domain to keep the mathematics sound. These exclusions often arise where the function becomes undefined. For a rational function \( h(x) = \frac{2}{(x+1)(x-4)} \), if you plug in a value of \( x \) that causes the denominator to be zero, the function loses its definition.• The core idea here is to identify which values of \( x \) those are, so we can avoid them. In this function:

• Setting \( x+1 \) to zero gives \( x = -1 \)
• and setting \( x-4 \) to zero gives \( x = 4 \).

These are the critical points at which the function is undefined because the denominator becomes zero.
Therefore, these values \( -1 \) and \( 4 \) are excluded from the domain. Ensuring these exclusions are clearly noted is crucial to accurately working with and understanding rational functions.