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Chapter 5: Problem 87

Think About It 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 k is therefore \(k = 3 / 12 = 0.25\)

Step by step solution

01

Substituting the value of the inverse function

For the given inverse function \(f^{-1}(3) = -2\), use this relation to substitute these values into the function, i.e., when the output is 3, the input is -2. So replace \(f(x)\) with 3 and \(x\) with -2 in the function, to get: \(3 = k \left(2 - (-2) - (-2)^3\right)\)
02

Simplifying the equation

Next, continue by simplifying the equation. This requires basic algebraic skills. The equation simplifies to \(3 = k \left(4 - (-8)\right)\) which implies \(3 = k \times 12\)
03

Solving for 'k'

Finally, solve the equation for 'k'. This is done by dividing both sides of the equation by 12, getting: \(k = 3 / 12\)

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Key Concepts

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

One-to-One Function
A one-to-one function, also known as an injective function, is a fundamental concept in calculus that ensures each element of the domain maps to a unique element in the range. In other words, if you input different numbers into a one-to-one function, you will always get out different results. This uniqueness is what allows for the existence of an inverse function, which we'll discuss in the next section.

Why does this matter in our exercise? Well, knowing that a function is one-to-one gives us the green light to look for its inverse, because it means that each output corresponds to one, and only one, input. This property is crucial when determining the constant 'k' using the inverse as stated in the problem.
Function Inverse
The inverse of a function reverses the function's operation. It swaps the roles of inputs and outputs, effectively undoing whatever the original function does. When you have a function 'f' and its inverse 'f-1', applying 'f' first and then 'f-1' will bring you back to your starting value. For the inverse to exist, the function must be one-to-one.

In the given exercise, the inverse function is used to find the constant 'k'. The relationship given by the inverse, that when the output is 3, the input must be -2, allows for a direct substitution in the equation. This gives an equation that can be manipulated, leading us to the value of 'k'.
Algebraic Manipulation
Algebraic manipulation is the process of reshaping an algebraic expression into a more usable form using the rules of arithmetic and algebra. This includes basic operations like addition, subtraction, multiplication, division, and the use of properties such as the distributive property to reorganize and simplify expressions.

In the context of our exercise, after substituting the inverse function values into the original function, algebraic manipulation is employed to simplify the resulting equation. This simplification, a key algebraic skill, gets us one step closer to isolating the constant 'k' and solving the problem.
Solving for Variables
Solving for variables is a central part of algebra and calculus. It involves finding the values of unknown quantities that satisfy a given equation. To solve for a variable, you must isolate it on one side of the equation, often using algebraic manipulation. The variable becomes the subject of the equation and can be calculated once it stands alone.

Step 3 of the solution illustrates solving for the variable 'k'. After simplifying the equation, we divide both sides by 12 to isolate 'k'. This straightforward process of isolating 'k' is essential in discovering its value, leading to the final solution of the exercise.

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