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Chapter 1: Problem 38

Evaluate the given function at the given values. \(f(x)=\frac{x-1}{x+1}, f(1), f(-1), f(x+1), f\left(x^{2}\right)\)

Short Answer

Expert verified
f(1) = 0, f(-1) is undefined, f(x+1) = \(\frac{x}{x+2}\), f(x^2) = \(\frac{x^2-1}{x^2+1}\)."

Step by step solution

01

Evaluate at f(1)

Substitute 1 for x in the function: \[ f(1) = \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0 \] Thus, \( f(1) = 0 \).
02

Evaluate at f(-1)

Substitute -1 for x in the function:\[ f(-1) = \frac{-1 - 1}{-1 + 1} = \frac{-2}{0} \]Since division by zero is undefined, \( f(-1) \) is undefined.
03

Evaluate at f(x+1)

Substitute \( x+1 \) for x in the function:\[ f(x+1) = \frac{(x+1) - 1}{(x+1) + 1} = \frac{x}{x+2} \]Thus, \( f(x+1) = \frac{x}{x+2} \).
04

Evaluate at f(x^2)

Substitute \( x^2 \) for x in the function:\[ f(x^2) = \frac{x^2 - 1}{x^2 + 1} \]Thus, \( f(x^2) = \frac{x^2 - 1}{x^2 + 1} \).

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Key Concepts

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

Substitution
Substitution is a fundamental concept in evaluating functions. It involves replacing the variable in the function with a given value or expression.
This process allows us to find the specific output for different inputs. For example, in evaluating the function \( f(x) = \frac{x-1}{x+1} \) at \( f(1) \), we replace \( x \) with \( 1 \). The equation becomes \( f(1) = \frac{1-1}{1+1} = \frac{0}{2} \).
Hence, the value of \( f(1) \) is 0.
  • Identify the value or expression to substitute.
  • Replace the variable with this value or expression in the function.
  • Perform the arithmetic operations to simplify the expression.
Substitution helps us understand how functions behave with specific inputs and is crucial for exploring concepts like undefined values and composite functions.
Undefined Value
An undefined value occurs when a mathematical expression does not result in a numerical output. This usually happens when a situation arises that is not permissible in mathematics. In functions, one typical cause of undefined values is division by zero.
In our given function \( f(x) = \frac{x-1}{x+1} \), while evaluating \( f(-1) \), we get \( \frac{-2}{0} \). Since dividing by zero is undefined, \( f(-1) \) cannot be calculated.
  • An undefined value often indicates a restriction or limitation within the function's domain.
  • Identifying undefined values helps us understand the boundaries or constraints of a function.
Understanding when a function is undefined will prevent mistakes when graphing or applying functions in various mathematical contexts.
Division by Zero
Division by zero is a fundamental rule in mathematics stating that dividing any number by zero is undefined or not possible. This is because dividing by zero would lead to an equation without a logical result or a number without end.
In the function \( f(x) = \frac{x-1}{x+1} \), trying \( f(-1) \) results in a division by zero, as the denominator becomes zero \( \frac{-2}{0} \).
  • It is important to evaluate expressions to check if division by zero occurs, which may prevent finding valid numerical solutions.
  • Functions should be defined to exclude values that cause the denominator to be zero.
Recognizing this concept helps prevent mistakes in calculations and ensures valid function results.
Composite Functions
Composite functions involve the substitution of one function into another. This is useful for evaluating complex expressions and understanding the interplay between different functions.
In our exercise, evaluating \( f(x+1) \) and \( f(x^2) \) showcases this idea:
  • For \( f(x+1) \), substitute \( x+1 \) for \( x \), resulting in the function \( \frac{x}{x+2} \), which changes the expression of the original function.
  • For \( f(x^2) \), substitute \( x^2 \) for \( x \), resulting in \( \frac{x^2 - 1}{x^2 + 1} \). This transforms the function's behavior for squared values of \( x \).
Composite functions help us explore how modifying inputs affect the function's output and can solve more complex real-world problems.

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Most popular questions from this chapter

Can you define \(\log _{a} x\) when \(a=1 ?\) Explain why or why not.

Inflation, as measured by Japan's consumer price index, \(^{65}\) decreased (thus the word deflation) by \(0.7 \%\) in the year \(2001 .\) If this rate were to continue for the next 10 years, use your computer or graphing calculator to determine how long before the value of a typical item would be reduced to \(95 \%\) of its value in 2001 .

Suppose the demand equation for a commodity is of the form \(p=m x+b\), where \(m<0\) and \(b>0\). Suppose the cost function is of the form \(C=d x+e\), where \(d>0\) and \(e<0 .\) Show that profit peaks before revenue peaks.

Crafton \(^{61}\) created a mathematical model of demand for northern cod and formulated the demand equation $$ p(x)=\frac{173213+0.2 x}{138570+x} $$ where \(p\) is the price in dollars and \(x\) is in kilograms. Graph this equation. Does the graph have the characteristics of a demand equation? Explain. Find \(p(0),\) and explain what the significance of this is.

Potts and Manooch \(^{71}\) studied the growth habits of coney groupers. These groupers are important components of the commercial fishery in the Caribbean. The mathematical model that they created was given by the equation \(L(t)=385(1-\) \(e^{-0.32[t-0.49]}\), where \(t\) is age in years and \(L\) is length in millimeters. Graph this equation. What seems to be happening to the length as the coneys become older? Potts and Manooch also created a mathematical model that connected length with weight and was given by the equation \(W(L)=2.59 \times 10^{-5} \cdot L^{2.94},\) where \(L\) is length in millimeters and \(W\) is weight in grams. Find the length of a 10 -year old coney. Find the weight of a 10 -year old coney.
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