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Chapter 1: Problem 2

Fill in the blanks.The inverse function of \(f\) is denoted by _______ .

Short Answer

Expert verified
The inverse function of \(f\) is denoted by \(f^{-1}\)

Step by step solution

01

Recall the concept of inverse functions

An inverse function is a function that 'reverses' another function. If the function \(f\) takes \(x\) to \(y\), then the inverse function takes \(y\) back to \(x\). The operation of the function is undone by its inverse.
02

Recall the notation for inverse functions

The inverse function of a function \(f\) is represented as \(f^{-1}\) in mathematical notation.

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Key Concepts

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

Function Notation
In mathematics, function notation is a way to denote functions clearly and succinctly. It allows us to represent functions using symbols, making it easier to write and work with them. Function notation is typically shown as:
  • If we have a function named "\(f\)", it is written as \(f(x)\), where \(x\) is the variable or input value.
  • The resulting value after applying \(f\) is called \(f(x)\), which is the output.

Function notation helps in linking inputs to outputs unambiguously. For example, if \(f(x) = x + 3\), then given \(x = 2\), the function's output would be noted as \(f(2) = 2 + 3 = 5\). This notation is fundamental when exploring further concepts like inverse functions, as it specifies clearly how inputs and outputs are transformed.
Inverse Operation
The concept of an inverse operation involves the process of reversing a function's action. Think of it as undoing the effect a function has on an input value.
  • For a function \(f\), if \(f(x)\) changes \(x\) into \(y\), the inverse operation will change \(y\) back into \(x\).
  • The inverse of a function is denoted as \(f^{-1}(y)\), and finding the inverse requires solving for \(x\) in terms of \(y\).

To validate an inverse operation, we should have \(f(f^{-1}(y)) = y\) and \(f^{-1}(f(x)) = x\). These equations confirm that applying a function and then its inverse returns us to our starting point. Inverse operations are crucial in algebra and calculus, especially in solving equations and understanding function behavior.
Mathematical Notation
Mathematical notation is the system of symbols used to represent numbers, operations, and concepts clearly. It's like a language that mathematicians use to communicate ideas and solve problems. In the case of functions, notation helps in expressing difficult concepts simply and consistently.
  • The notation \(f^{-1}\) signifies the inverse of function \(f\). This symbol doesn't mean to raise \(f\) to the power of -1 but instead indicates a function that undoes \(f\).

  • Proper notation ensures that mathematical expressions are universally understood, meaning the same thing every time they are used.

Consider simple expressions like \(a + b\) or complex ones like the integral \(\int f(x)\,dx\). Once we learn what the symbols mean, we can read and write a vast range of mathematical expressions. Expanding our understanding of mathematical notation is essential for advancing in diverse fields of mathematics.

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Most popular questions from this chapter

True or False? Determine whether the statement is true or false. Justify your answer. It is possible for an odd function to have the interval \([0, \infty)\) as its domain.

Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.

It The function \(f(x)=k\left(2-x-x^{3}\right)\) Thas an inverse function, and \(f^{-1}(3)=-2 .\) Find \(k\).

The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the driver's reaction time is given by \(R(x)=\frac{3}{4} x,\) where \(x\) is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given by \(B(x)=\frac{1}{15} x^{2}.\) (a) Find the function that represents the total stopping distance \(T.\) (b) Graph the functions \(R, B,\) and \(T\) on the same set of coordinate axes for \(0 \leq x \leq 60.\) (c) Which function contributes most to the magnitude of the sum at higher speeds? Explain.

Find a mathematical model for the verbal statement. The rate of growth \(R\) of a population is jointly proportional to the size \(S\) of the population and the difference between \(S\) and the maximum population size \(L\) that the environment can support.
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