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Chapter 1: Problem 83

Determine whether 1 is in the range of \(f(x)=\frac{x-1}{x+1}\)

Short Answer

Expert verified
So, 1 is not in the range of the given function \(f(x) = \frac{x-1}{x+1}\)

Step by step solution

01

Set the Function Equal to 1

Firstly, set the given function equal to 1, as follows: \(f(x) = 1\). Substituting \(f(x) = \frac{x-1}{x+1}\) we get \(\frac{x-1}{x+1} = 1\).
02

Solve for x

Next, solve the equation derived in Step 1, for x. Cross multiply to remove the fraction : \(x - 1 = x + 1\)
03

Simplify

Simplify the equation to isolate x. By subtracting x from both sides of the equation, and then adding 1 to both sides we get, 0 = 2. This is clearly not possible, so there is no value of x satisfying the original equation.

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Key Concepts

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

Solving Rational Equations
Rational equations are equations that feature rational expressions, which are fractions with polynomials in the numerator, denominator, or both. When faced with such equations, the main goal is to eliminate the fractions. A common technique involves what is known as cross-multiplication. This means multiplying both sides of an equation by the denominator to remove the fraction.
  • Step 1: Determine the equation or expression and identify the fractions that need resolving.
  • Step 2: If necessary, rearrange the equation so that there is a fraction on each side. This makes cross-multiplying straightforward.
  • Step 3: Perform cross-multiplication to clear the fractions.
  • Step 4: Solve the resulting polynomial equation.
In our case of the function \( f(x) = \frac{x-1}{x+1} \), setting it equal to 1, and cross-multiplying, ensured we dealt directly with equivalence, allowing us to move forward neatly.
Function Equality
Function equality occurs when two expressions are equal across their domains. To determine if specific elements belong to the range of a function, we test whether the output value can be obtained by any input. This involves setting the function equal to the desired range value.
  • Step 1: Identify the desired output value you want to check for within the function's range.
  • Step 2: Set the function expression equal to this value.
  • Step 3: Solve for the variable to determine if there is any input value that gives the output.
For \( f(x) = \frac{x-1}{x+1} \), setting \( f(x) = 1 \) simulates checking if 1 is indeed an available output for any input. This process gives insight into its range without checking every possible x-value.
Algebraic Manipulation
Algebraic manipulation refers to rearranging and simplifying expressions to find solutions to equations. These techniques can include distributing terms, factoring, expanding, or combining like terms.
  • Start by rearranging the equation to isolate terms of interest.
  • Simplify both sides of the equation by combining like terms.
  • Perform operations such as addition or subtraction to further isolate the variables.
  • Check your solution to verify its correctness.
In the example, after setting the equation \( \frac{x-1}{x+1} = 1 \), algebraic manipulation was used to simplify to \( x - 1 = x + 1 \). The process sought a value of x that works; however, after eliminating x from both sides, the resulting false statement \( 0 = 2 \) indicated no solutions, affirming 1 is not in the function's range.

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