Problem 108
Question
Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\begin{aligned}&\text { Let } f(x)=\frac{5}{2 x-4} \text { and } g(x)=\frac{3}{2-x}\\\ &\text { Find } f(x)-g(x)\end{aligned}$$
Step-by-Step Solution
Verified Answer
\(-\frac{1}{2(x-2)}\)
1Step 1: Identify the Expressions
We begin by identifying the rational expressions for the functions. For \( f(x) \), we have \( \frac{5}{2x-4} \), and for \( g(x) \), we have \( \frac{3}{2-x} \). We want to find \( f(x) - g(x) \).
2Step 2: Rewrite the Denominator
Notice that \( 2-x \) can be rewritten as \( -(x-2) \). Thus, the expression \( \frac{3}{2-x} \) is equivalent to \( -\frac{3}{x-2} \).
3Step 3: Identify a Common Denominator
The denominators \( 2x-4 \) and \( x-2 \) can be rewritten for commonality. Factor \( 2x-4 \) as \( 2(x-2) \). Hence, our expressions become \( \frac{5}{2(x-2)} \) and \( -\frac{3}{x-2} \). A common denominator for \( \frac{5}{2(x-2)} \) and \( -\frac{3}{x-2} \) is \( 2(x-2) \).
4Step 4: Rewrite Expressions with Common Denominator
Rewrite each expression to have the common denominator \( 2(x-2) \): - The first expression, \( \frac{5}{2(x-2)} \), is already in terms of the common denominator.- Rewrite the second expression, \( -\frac{3}{x-2} \), by multiplying the numerator and the denominator by 2 to get \( -\frac{6}{2(x-2)} \).
5Step 5: Subtract the Expressions
Now, we subtract the two rational expressions: \[ \frac{5}{2(x-2)} - \frac{6}{2(x-2)} = \frac{5 - 6}{2(x-2)} \]
6Step 6: Simplify the Resulting Expression
Simplify the numerator in the resulting expression: \[ \frac{5 - 6}{2(x-2)} = \frac{-1}{2(x-2)} \] Thus, the simplified result of \( f(x) - g(x) \) is \( \frac{-1}{2(x-2)} \).
Key Concepts
Addition and Subtraction of Rational ExpressionsCommon DenominatorSimplifying Rational ExpressionsFunction Operations
Addition and Subtraction of Rational Expressions
When dealing with rational expressions, think of them like fractions with variables. Just like adding or subtracting fractions, you need to find a common denominator. This means both expressions must have the same base in their denominators.
Only then can you add or subtract the numerators directly. This ensures the expressions are compatible. This step is crucial because it sets the foundation for simplifying within the same terms. Here, when subtracting \( f(x) - g(x) \), we're finding the difference of two rational expressions, and the same logic applies as with simple fractions.
It's essential to handle operations on the numerators as if you're working with regular numbers, keeping a keen eye on signs (positive or negative). Pay special attention to their signs. They profoundly affect the final outcome.
Only then can you add or subtract the numerators directly. This ensures the expressions are compatible. This step is crucial because it sets the foundation for simplifying within the same terms. Here, when subtracting \( f(x) - g(x) \), we're finding the difference of two rational expressions, and the same logic applies as with simple fractions.
It's essential to handle operations on the numerators as if you're working with regular numbers, keeping a keen eye on signs (positive or negative). Pay special attention to their signs. They profoundly affect the final outcome.
Common Denominator
A common denominator is a shared base of two or more denominators in a rational expressions operation.
Without it, we can't add or subtract these expressions accurately. To find a common denominator, first look to rewrite any denominators. In our example, the denominators \(2x-4\) and \(2-x\), can be rewritten.
Notice "\(2-x\)" is the same as "\(-(x-2)\)". We can factor \(2x-4\) as \(2(x-2)\). This simple step is key in transforming our problem into something solvable. Hence we establish \(2(x-2)\) as the common denominator. Once equal, the expressions become compatible for further operations.
Without it, we can't add or subtract these expressions accurately. To find a common denominator, first look to rewrite any denominators. In our example, the denominators \(2x-4\) and \(2-x\), can be rewritten.
Notice "\(2-x\)" is the same as "\(-(x-2)\)". We can factor \(2x-4\) as \(2(x-2)\). This simple step is key in transforming our problem into something solvable. Hence we establish \(2(x-2)\) as the common denominator. Once equal, the expressions become compatible for further operations.
Simplifying Rational Expressions
After achieving a common denominator, simplifying rational expressions becomes straightforward. In our calculation, \( \frac{5}{2(x-2)} - \frac{6}{2(x-2)} \) was simplified by subtracting numerators, as they share a common denominator.
The numerator, \( 5-6 \), becomes \( -1 \) upon simplification. The expression thereby reduces to \( \frac{-1}{2(x-2)} \).
Simplifying is about reducing expressions to their simplest form. It makes them easy for further mathematical operations or understanding the expression's nature. Double-check if you can further simplify any numeric or variable parts. However, in some cases like ours, what you have is already the simplest form possible.
The numerator, \( 5-6 \), becomes \( -1 \) upon simplification. The expression thereby reduces to \( \frac{-1}{2(x-2)} \).
Simplifying is about reducing expressions to their simplest form. It makes them easy for further mathematical operations or understanding the expression's nature. Double-check if you can further simplify any numeric or variable parts. However, in some cases like ours, what you have is already the simplest form possible.
Function Operations
Managing functions like \( f(x) \) and \( g(x) \) particularly involves handling additions or subtractions through their respective expressions.
In this scenario, combine \( f(x) \) and \( g(x) \) by taking the difference of their given rational forms. Each function is a separate entity, but operations can be performed between them seamlessly once their expressions align.
Components like \( x \) are variables that provide a range of values upon substitution, allowing the function to yield results. It's vital because operations on functions extend beyond just manipulation; they represent transformations and relationships. Grasping function operations offers a deeper understanding of how different algebraic expressions interact in complex mathematical contexts.
In this scenario, combine \( f(x) \) and \( g(x) \) by taking the difference of their given rational forms. Each function is a separate entity, but operations can be performed between them seamlessly once their expressions align.
Components like \( x \) are variables that provide a range of values upon substitution, allowing the function to yield results. It's vital because operations on functions extend beyond just manipulation; they represent transformations and relationships. Grasping function operations offers a deeper understanding of how different algebraic expressions interact in complex mathematical contexts.
Other exercises in this chapter
Problem 108
Let \(f(x)=\frac{x^{3}+2 x^{2}-32}{x} .\) For what values of \(x\) is \(f(x)=16 ?\)
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Perform the indicated operations. $$ \left(5 k-6 m^{2}\right)^{2} $$
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Perform each operation. $$ \left(4 u^{2}+z^{2}-3 u^{2} z^{2}\right)-\left(u^{3}+3 z^{2}-3 u^{2} z^{2}\right) $$
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Let \(f(x)=\frac{2 x^{3}+x^{2}}{98 x+49} .\) For what values of \(x\) is \(f(x)=1 ?\)
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