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Magazine Chapter 8: Problem 14
If \(f(x)=\frac{1}{x^{2}},\) find \(-f(x)+2 f(x-3) ;\) then find a common denominator and combine into one rational expression.
Short Answer
Expert verified
\( \frac{x^2 + 6x - 9}{x^2 (x-3)^2} \)
Step by step solution
01
- Express the given function
Given the function is: \[ f(x) = \frac{1}{x^2} \]
02
- Calculate \( -f(x) \)
To find \( -f(x) \), we multiply the function by -1: \[ -f(x) = - \frac{1}{x^2} \]
03
- Calculate \( 2f(x-3) \)
To find \( 2f(x-3) \), we substitute \( x-3 \) into the function and then multiply by 2: \[ f(x-3) = \frac{1}{(x-3)^2} \] \[ 2f(x-3) = 2 \cdot \frac{1}{(x-3)^2} = \frac{2}{(x-3)^2} \]
04
- Combine the expressions \(-f(x) + 2f(x-3)\)
Now, we combine \(-f(x)\) and \(2f(x-3)\): \[ -f(x) + 2f(x-3) = - \frac{1}{x^2} + \frac{2}{(x-3)^2} \]
05
- Find a common denominator
To combine the fractions, find the common denominator: \[ \text{LCM}(x^2, (x-3)^2) = x^2 (x-3)^2 \] \[ - \frac{1}{x^2} = - \frac{(x-3)^2}{x^2 (x-3)^2} \] \[ \frac{2}{(x-3)^2} = \frac{2x^2}{x^2 (x-3)^2} \]
06
- Combine into a single rational expression
Combine the terms over the common denominator: \[ - \frac{(x-3)^2}{x^2 (x-3)^2} + \frac{2x^2}{x^2 (x-3)^2} = \frac{- (x-3)^2 + 2x^2}{x^2 (x-3)^2} \]
07
- Simplify the numerator
Expand \( (x-3)^2 \) and then simplify the numerator: \[ (x-3)^2 = x^2 - 6x + 9 \] \[ - (x-3)^2 + 2x^2 = - x^2 + 6x - 9 + 2x^2 = x^2 + 6x - 9 \]
08
- Write the final expression
Combine everything to get the final answer: \[ \frac{x^2 + 6x - 9}{x^2 (x-3)^2} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combining Fractions
Combining fractions means adding or subtracting them to create a single fraction. To do this, the fractions must share a common denominator. This ensures the fractions are on the same base and can be combined easily.
For example, combining \(-\frac{1}{x^2}\) and \(2 \frac{1}{(x-3)^2}\) involves transforming both fractions to have a shared denominator before performing addition or subtraction.
Once the denominators are the same, you simply add or subtract the numerators, while keeping the common denominator unchanged.
For example, combining \(-\frac{1}{x^2}\) and \(2 \frac{1}{(x-3)^2}\) involves transforming both fractions to have a shared denominator before performing addition or subtraction.
Once the denominators are the same, you simply add or subtract the numerators, while keeping the common denominator unchanged.
Common Denominator
A common denominator is shared by two or more fractions, allowing us to combine them. To find a common denominator, we often use the Least Common Multiple (LCM) of the current denominators. The LCM is the smallest number that is a multiple of each of the denominators.
In our exercise, the denominators are \(x^2\) and \((x-3)^2\). The common denominator is found by multiplying these together: \(x^2 (x-3)^2\). This turns both fractions into equivalent fractions with the same base:
\(-\frac{1}{x^2} = -\frac{(x-3)^2}{x^2 (x-3)^2}\)
\(\frac{2}{(x-3)^2} = \frac{2x^2}{x^2 (x-3)^2}\).
This allows for easier combination of the numerators.
In our exercise, the denominators are \(x^2\) and \((x-3)^2\). The common denominator is found by multiplying these together: \(x^2 (x-3)^2\). This turns both fractions into equivalent fractions with the same base:
\(-\frac{1}{x^2} = -\frac{(x-3)^2}{x^2 (x-3)^2}\)
\(\frac{2}{(x-3)^2} = \frac{2x^2}{x^2 (x-3)^2}\).
This allows for easier combination of the numerators.
Function Transformation
Function transformation involves shifting or changing a function in specific ways, such as translating, scaling, or reflecting it.
In this exercise, we transformed the function by substituting \(x-3\) in place of \(x\) to get \(f(x-3)\). This shifts the function to the right by 3 units. Applying a multiplier like 2 to \(f(x-3)\) scales the function vertically:
Replacing the variable:
\(f(x-3)=\frac{1}{(x-3)^2}\)
Scaling the function:
\(2f(x-3)=2\frac{1}{(x-3)^2}=\frac{2}{(x-3)^2}\).
The negative sign applied to \(f(x)\) reflects the function across the x-axis.
In this exercise, we transformed the function by substituting \(x-3\) in place of \(x\) to get \(f(x-3)\). This shifts the function to the right by 3 units. Applying a multiplier like 2 to \(f(x-3)\) scales the function vertically:
Replacing the variable:
\(f(x-3)=\frac{1}{(x-3)^2}\)
Scaling the function:
\(2f(x-3)=2\frac{1}{(x-3)^2}=\frac{2}{(x-3)^2}\).
The negative sign applied to \(f(x)\) reflects the function across the x-axis.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing fractions to their simplest form, where numerator and denominator have no common factors other than 1.
In this exercise, we combined the fractions with a common denominator and then simplified the numerator:
1. Expand \((x-3)^2 = x^2 - 6x + 9\)
2. Combine the expanded term with \(-x^2\) and \(2x^2\)
3. Simplify the numerator expression \(- (x-3)^2 + 2x^2 = x^2 + 6x - 9\)
Resulting in:
\(\frac{x^2 + 6x - 9}{x^2 (x-3)^2}\)
This process reduces the complexity of the expression, making it easier to understand and work with.
In this exercise, we combined the fractions with a common denominator and then simplified the numerator:
1. Expand \((x-3)^2 = x^2 - 6x + 9\)
2. Combine the expanded term with \(-x^2\) and \(2x^2\)
3. Simplify the numerator expression \(- (x-3)^2 + 2x^2 = x^2 + 6x - 9\)
Resulting in:
\(\frac{x^2 + 6x - 9}{x^2 (x-3)^2}\)
This process reduces the complexity of the expression, making it easier to understand and work with.
