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Magazine Chapter 5: Problem 31
\(f(x)=\frac{(x+1)^{2}}{x^{2}+1}\)
Short Answer
Expert verified
1 + \frac{2x}{x^2 + 1}
Step by step solution
01
Identify the function
The given function is \( f(x) = \frac{(x+1)^2}{x^2+1} \).
02
Simplify the numerator
Expand the numerator: \( (x+1)^2 = x^2 + 2x + 1 \).
03
Simplify the function
Substitute the expanded numerator back into the function: \( f(x) = \frac{x^2 + 2x + 1}{x^2 + 1} \).
04
Split the function into parts
Express the function as the sum of separate fractions: \( f(x) = \frac{x^2 + 2x + 1}{x^2 + 1} = \frac{x^2 + 1 + 2x}{x^2 + 1} = \frac{x^2 + 1}{x^2 + 1} + \frac{2x}{x^2 + 1} \).
05
Simplify each part
Simplify the first part: \( \frac{x^2 + 1}{x^2 + 1} = 1 \). Simplify the second part: \( \frac{2x}{x^2 + 1} \).
06
Combine the simplified parts
Combine the results from the simplified parts: \( f(x) = 1 + \frac{2x}{x^2 + 1} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
rational functions
A rational function is a fraction where both the numerator and the denominator are polynomials. The given exercise features the function \( f(x) = \frac{(x+1)^2}{x^2+1} \).
Rational functions are vital in algebra because they appear in many real-world applications, from physics to economics. To analyze and simplify rational functions, you need to understand polynomial arithmetic.
Rational functions are vital in algebra because they appear in many real-world applications, from physics to economics. To analyze and simplify rational functions, you need to understand polynomial arithmetic.
- The function's behavior is influenced by both the numerator and the denominator.
- To simplify, start by examining the degrees of the polynomials in the numerator and the denominator.
- Consider if factoring or other algebraic techniques can rewrite the function in a more manageable form.
algebraic manipulation
Algebraic manipulation involves rewriting expressions in more useful or simplified forms using algebraic rules and operations. For rational functions, this might include expanding polynomials, factoring, or combining fractions.
In our example, we expanded \( (x+1)^2 \) to \( x^2 + 2x + 1 \). This expansion lets us see the terms clearly and helps simplify further steps.
In our example, we expanded \( (x+1)^2 \) to \( x^2 + 2x + 1 \). This expansion lets us see the terms clearly and helps simplify further steps.
- Always use distributive property to expand expressions as needed.
- Combine like terms meticulously to avoid mistakes.
- After manipulating the algebraic expressions, re-simplify the result to ensure correctness.
numerator and denominator simplification
Often, simplifying a rational function focuses on making the numerator and denominator as simple as possible. This process affects how we can work with the whole function.
For the function \( f(x) = \frac{(x+1)^2}{x^2+1} \), expanding the numerator helped reveal a form that makes simplifying the whole function easier: \( f(x) = \frac{x^2 + 2x + 1}{x^2 + 1} \).
By writing it as a sum of fractions: \( f(x) = \frac{x^2 +1 + 2x}{x^2 + 1} = \frac{x^2 + 1}{x^2 + 1} + \frac{2x}{x^2 + 1} \), we see parts of the expression simplified further: \( 1 + \frac{2x}{x^2 + 1} \).
For the function \( f(x) = \frac{(x+1)^2}{x^2+1} \), expanding the numerator helped reveal a form that makes simplifying the whole function easier: \( f(x) = \frac{x^2 + 2x + 1}{x^2 + 1} \).
By writing it as a sum of fractions: \( f(x) = \frac{x^2 +1 + 2x}{x^2 + 1} = \frac{x^2 + 1}{x^2 + 1} + \frac{2x}{x^2 + 1} \), we see parts of the expression simplified further: \( 1 + \frac{2x}{x^2 + 1} \).
- Factor or expand where necessary for clearer analysis.
- Identify and cancel out common factors in numerator and denominator, if available.
- Always verify simplifications by substituting values or alternative checks.
partial fraction decomposition
Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions. While our example didn't require extensive use of this technique, it lays the groundwork for its necessity in more complicated functions.
Here鈥檚 how it works:
Mastery of partial fraction techniques is useful for integrals in calculus and solving differential equations.
Here鈥檚 how it works:
- First, ensure the polynomial in the numerator is of lower degree than in the denominator. If not, perform polynomial long division.
- Next, factor the denominator if possible.
- Then, express the function as a sum of simpler fractions.
- Finally, solve for the unknown coefficients.
Mastery of partial fraction techniques is useful for integrals in calculus and solving differential equations.
