/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 95 The function \(f(x)=k\left(2-x-x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 95

The function \(f(x)=k\left(2-x-x^{3}\right)\) is one-to-one and \(f^{-1}(3)=-2\). Find \(k\).

Short Answer

Expert verified
The value of the constant \(k\) in the given function is 0.25.

Step by step solution

01

Substitute the Inverse Function Value

Substitute the given \(f^{-1}(3)\) value in the function. In this case, \(f^{-1}(3)=-2\), so we substitute -2 into the function: \(f(-2)\). This would give \(f(-2) = k \left(2 - (-2) - (-2)^3\right)\)
02

Simplify the Expression

Simplify the equation from step 1. &nbs This will give us \(f(-2) = k \left(2+2-(-8)\)\), which simplifies to \(f(-2) = 12k\).
03

Use the Property of the Inverse Function

According to the property of the inverse functions, \(f(f^{-1}(a))=a\) for any number \(a\). Our function \(f(-2)\) equals to 3. So, by equating the two expressions we get \(12k = 3\)
04

Solve for \(k\)

Solving `12k = 3` for \(k\), will give the value of \(k\). It then shall be seen that \(k=3/12\), which simplifies to \(k=0.25\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

One-to-One Functions
A one-to-one function, also called an injective function, is a type of function where each element in the domain maps to a unique element in the codomain. This means that no two different inputs result in the same output. Understanding whether a function is one-to-one is essential, especially when dealing with inverse functions, because only one-to-one functions have inverses.
  • This unique mapping allows you to "reverse" the function, finding the original input from its output, which is the heart of inverse functions.
  • To check if a function is one-to-one, ensure that for every pair of different inputs, the outputs are also different.
For example, if you have a function where you input two different numbers and always get different results, it is one-to-one. For the function given in the exercise, it is already mentioned to be one-to-one, meaning it has an inverse function.
Inverse Function Property
The inverse function property is a fundamental characteristic in math that provides a relationship between a function and its inverse. If a function \( f \) has an inverse \( f^{-1} \), then for any number \( a \), the property \( f(f^{-1}(a)) = a \) holds true. This means that if you apply a function to its inverse, you should retrieve the original input value.
  • This property guarantees the unique mapping characteristic of a one-to-one function where the function \( f \) with its inverse \( f^{-1} \) "cancels out," returning the original input.
  • In our exercise, for instance, \( f(f^{-1}(3)) = 3 \), since \( f^{-1}(3) = -2 \).
This relationship was used in the step-by-step solution to find the value of \( k \). By setting \( f(-2) = 3 \) and using the inverse property, we can solve for unknown values.
Solving Equations
Solving equations involves finding the value of the unknown variable that makes the equation true. In our exercise, after using the inverse function property, we have an equation: \( 12k = 3 \). Solving this involves simple algebraic manipulation to isolate \( k \).
  • First, divide both sides of the equation by 12, which simplifies the expression. Therefore, \( k = \frac{3}{12} \).
  • Further simplification gives us \( k = 0.25 \).
Equations like these are common in problems involving inverse functions. Solving them accurately is crucial for finding correct values, as seen in our solution where we derived \( k \). Remember to check your work by substituting back into the original equation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the limit. $$ \lim _{x \rightarrow 0} \frac{\sinh x}{x} $$

From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at a point \(P\). Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P\).

Find the integral. $$ \int \frac{\cosh x}{\sinh x} d x $$

Find the derivative of the function. $$ y=\sinh ^{-1}(\tan x) $$

Prove or disprove that there is at least one straight line normal to the graph of \(y=\cosh x\) at a point \((a, \cosh a)\) and also normal to the graph of \(y=\sinh x\) at a point \((c, \sinh c)\). [At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, \(\cosh x=\left(e^{x}+e^{-x}\right) / 2\) and \(\left.\sinh x=\left(e^{x}-e^{-x}\right) / 2 .\right]\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.