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Magazine Chapter 6: Problem 42
If \(f(x)=\frac{x-1}{x+1}, x \neq-1 ;\) then find \(f\left(\frac{x-1}{x+1}\right)\). (1) \(x\) (2) \(-\frac{1}{\mathrm{x}}\) (3) \(\mathrm{f}(\mathrm{x})\) (4) \(\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)\)
Short Answer
Expert verified
Answer: \(f\left(\frac{x-1}{x+1}\right)=-\frac{1}{x}\)
Step by step solution
01
Write down the given function definition
We're given the function, \(f(x)=\frac{x-1}{x+1}\).
02
Substitute \(\frac{x-1}{x+1}\) in place of x
We have to find \(f\left(\frac{x-1}{x+1}\right)\), so we will substitute \(\frac{x-1}{x+1}\) in place of \(x\) in the given function:
\(f\left(\frac{x-1}{x+1}\right)=\frac{\frac{x-1}{x+1}-1}{\frac{x-1}{x+1}+1}\)
03
Simplify the numerator
To simplify the numerator, find a common denominator and combine like terms:
\(\frac{x-1}{x+1}-1=\frac{x-1-x-1}{x+1}=\frac{-2}{x+1}\)
04
Simplify the denominator
To simplify the denominator, find a common denominator and combine like terms:
\(\frac{x-1}{x+1}+1=\frac{x-1+x+1}{x+1}=\frac{2x}{x+1}\)
05
Substitute the simplified numerator and denominator
Now we substitute the simplified numerator and denominator back into our original expression:
\(f\left(\frac{x-1}{x+1}\right)=\frac{\frac{-2}{x+1}}{\frac{2x}{x+1}}\)
06
Simplify the function value
To simplify this expression, multiply the numerator and denominator by the least common denominator \((x+1)\):
\(f\left(\frac{x-1}{x+1}\right)=\frac{-2}{2x}\)
07
Simplify further and find the matching option
Divide the numerator and denominator by 2 to get the final result:
\(f\left(\frac{x-1}{x+1}\right)=-\frac{1}{x}\)
Comparing this result with the given options, we see that our answer matches option (2). So, \(f\left(\frac{x-1}{x+1}\right)=-\frac{1}{x}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Substitution
Function substitution is a powerful tool when dealing with complex functions or equations. In this problem, our task was to find the value of the function \(f(x)\) when a new expression is placed in place of \(x\). The given function was \(f(x)=\frac{x-1}{x+1}\). Our goal was to find \(f\left(\frac{x-1}{x+1}\right)\).
- To achieve this, we substitute \(\frac{x-1}{x+1}\) wherever we see \(x\) in the function \(f(x)\).
- This substitution changes the original equation to a more complex expression: \(f\left(\frac{x-1}{x+1}\right)=\frac{\frac{x-1}{x+1}-1}{\frac{x-1}{x+1}+1}\).
Fraction Simplification
Fraction simplification involves reducing the complexity of fractions to their simplest form. When we replaced \(x\) in the function with \(\frac{x-1}{x+1}\), the resulting expression was complex. It required us to simplify two parts of the fraction:
- Numerator simplification: We simplified the numerator \(\frac{x-1}{x+1}-1\) as follows. First, rewrite \(-1\) as \(\frac{-x-1}{x+1}\) to have a common denominator, resulting in: \(\frac{x-1-x-1}{x+1} = \frac{-2}{x+1}\).
- Denominator simplification: Similarly, the denominator \(\frac{x-1}{x+1}+1\) was simplified by rewriting \(1\) as \(\frac{x+1}{x+1}\), leading to: \(\frac{x-1+x+1}{x+1} = \frac{2x}{x+1}\).
Mathematical Problem Solving
Solving mathematical problems involves breaking down the problem into manageable steps, using logic and known mathematical techniques. In this exercise, we used systematic problem-solving approaches:
- Understanding the problem: First, we interpreted the given function and identified the substitutions required. The objective was clear: find \(f\left(\frac{x-1}{x+1}\right)\).
- Step-by-step calculations: We executed the function substitution and simplified the resulting fractions through deliberate steps, which are critical in ensuring accuracy.
- Selecting the correct answer: Finally, by comparing our simplified solution \(\frac{-1}{x}\) against the multiple choices provided, we verified that option (2) was correct.
