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Chapter 5: Problem 17

Find the domain of each rational function. $$ H(x)=\frac{-4 x^{2}}{(x-2)(x+4)} $$

Short Answer

Expert verified
Domain: \( x eq 2 \) and \( x eq -4 \)

Step by step solution

01

- Identify the denominator

The domain of a rational function is restricted by values that make the denominator equal to zero. For the given function, identify the denominator: \[ (x-2)(x+4) \]
02

- Set the denominator to zero

Set the denominator equal to zero to find the values of x that are not in the domain: \[ (x-2)(x+4) = 0 \]
03

- Solve the equation

Solve the equation from Step 2 by setting each factor to zero individually:\[ x-2 = 0 \] so \[ x = 2 \] and\[ x+4 = 0 \] so \[ x = -4 \]
04

- State the domain

The domain of the function consists of all real numbers except the values that make the denominator zero. Therefore, the domain is: \[ x eq 2 \] and \[ x eq -4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Denominator
In a rational function, the denominator is a vital part to consider. The denominator is the bottom part of the fraction that cannot be zero because division by zero is undefined. For the given function \( H(x)=\frac{-4 x^{2}}{(x-2)(x+4)} \), the denominator is \( (x-2)(x+4) \). Identifying the denominator is the first step to finding the domain of the function because it helps pinpoint the values of x that must be excluded. Always remember, when working with rational functions, the values making the denominator zero are crucial points to check.
Excluded Values
Excluded values in a rational function are specific values of x that make the denominator equal to zero. To determine these, we set the denominator equal to zero and solve for x. For \( H(x)=\frac{-4 x^{2}}{(x-2)(x+4)} \), we set \( (x-2)(x+4) = 0 \). Solving this, we get \( x-2=0 \) and \( x+4=0 \), which leads to \( x=2 \) and \( x=-4 \). These are the values that make the denominator zero. Therefore, the function is undefined at \( x=2 \) and \( x=-4 \), and these values are excluded from the domain. The domain of the function is consequently all real numbers except 2 and -4.
Factoring
Factoring is a useful technique when working with rational functions. It simplifies the process of finding excluded values by breaking down the denominator into its factors. In \( H(x)=\frac{-4 x^{2}}{(x-2)(x+4)} \), the denominator \( (x-2)(x+4) \) is already factored. This makes it easier to identify the values of x that make each factor zero, helping us find the excluded values more efficiently. Factoring transforms complex expressions into simpler parts, aiding in the overall understanding and solving of rational functions. Whenever you see a rational function, check if the denominator can be factored to simplify your work.

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