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Chapter 2: Problem 51

Find the inverse function of \(f.\) $$f(x)=5-4 x^{3}$$

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = \sqrt[3]{\frac{5 - x}{4}}\).

Step by step solution

01

Validate the Function

Ensure that the function is one-to-one by checking its derivative. The function is given by \(f(x) = 5 - 4x^3\). Its derivative \(f'(x)\) is \(-12x^2\). Since \(-12x^2\) is either negative or zero for all real \(x\), the function is strictly decreasing, indicating it is one-to-one.
02

Replace Function Notation with Variables

Set \(y = f(x)\), making the equation \(y = 5 - 4x^3\). This equation will be used to find \(x\) in terms of \(y\).
03

Solve for x

Rearrange the equation to solve for \(x\). Start with subtracting 5 from both sides: \(y - 5 = -4x^3\). Divide both sides by \(-4\) to isolate \(x^3\): \(x^3 = \frac{5 - y}{4}\).
04

Obtain x by Taking the Cube Root

Take the cube root of both sides: \(x = \sqrt[3]{\frac{5 - y}{4}}\). This expression allows us to express \(x\) in terms of \(y\).
05

Determine the Inverse Function

Write the inverse function by switching \(x\) and \(y\), ending with \(f^{-1}(x) = \sqrt[3]{\frac{5 - x}{4}}\).

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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 is a function where each output is uniquely mapped from one input, and no two different inputs map to the same output. This is an essential characteristic when finding inverse functions because only one-to-one functions have inverses that are also functions.
  • **Identification:** Typically, you identify a one-to-one function by ensuring that every horizontal line intersects the graph of the function at most once. This is known as the Horizontal Line Test.
  • **Example:** In the given function, \(f(x) = 5 - 4x^3\), the derivative helps confirm it is one-to-one.
By evaluating the derivative of \(f(x)\), which is \(f'(x) = -12x^2\), you notice that for all real x, the values are non-positive. A negative or zero derivative implies a non-increasing function, solidifying the one-to-one nature of \(f(x)\). This is crucial to ensuring the existence of an inverse that itself behaves as a function.
The Derivative
The derivative of a function expresses the rate at which the function's output value changes as the input changes. It's a core concept in calculus that aids in understanding behavior such as increasing or decreasing trends of the function.
In our exercise, we look at the derivative of \(f(x) = 5 - 4x^3\). Calculating the derivative yields \(f'(x) = -12x^2\). This tells us a couple of important things:
  • Since all squared numbers \(x^2\) are non-negative, \(-12x^2\) will always be non-positive. This confirms the function is either constant or decreasing.
  • A strictly decreasing function, confirmed by the derivative being always negative or zero, ensures it passes the criteria for being one-to-one.
Thus, through the derivative, we affirm this function is eligible to have an inverse, simplifying the path to finding \((f^{-1}(x))\).
Cubic Equations
Cubic equations, which involve expressions in the form of \(ax^3 + bx^2 + cx + d = 0\), can have up to three real roots. The shape of their graphs is a smooth curve that can have one inflection point. Understanding their form is important when performing operations like finding inverses.
In our problem, the given function is a cubic equation \(f(x) = 5 - 4x^3\). Solving this equation for \(x\) involved reversing the operations applied to \(x\)
  • Starting by isolating terms: rearrange \(-4x^3 = y - 5\).
  • Then, dividing by coefficients to isolate the cubic term: \(x^3 = \frac{5 - y}{4}\).
  • Finally, applying cube root to find \(x\): \(x = \sqrt[3]{\frac{5-y}{4}}\).
This reverses the initial cubic function to provide its inverse. Understanding cubic equations allows the transformation from one form to another, crucial when deriving the inverse precisely.

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Most popular questions from this chapter

When a bowl of hot soup is left in a room, the soup eventually cools down to room temperature. The temperature \(T\) of the soup is a function of time \(t .\) The table below gives the temperature (in "F) of a bowl of soup \(t\) minutes after it was set on the table. Find the average rate of change of the temperature of the soup over the first 20 minutes and over the next 20 minutes. During which interval did the soup cool off more quickly? $$\begin{array}{|c|c||c|c|} \hline t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) & t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) \\ \hline 0 & 200 & 35 & 94 \\ 5 & 172 & 40 & 89 \\ 10 & 150 & 50 & 81 \\ 15 & 133 & 60 & 77 \\ 20 & 119 & 90 & 72 \\ 25 & 108 & 120 & 70 \\ 30 & 100 & 150 & 70 \\ \hline \end{array}$$

Graphing Functions Sketch a graph of the function by first making a table of values. $$H(x)=|2 x|$$

An object is dropped from a high cliff, and the distance (in feet) it has fallen after \(t\) seconds is given by the function \(d(t)=16 t^{2}\) Complete the table to find the average speed during the given time intervals. Use the table to determine what value the average speed approaches as the time intervals get smaller and smaller. Is it reasonable to say that this value is the speed of the object at the instant \(t=3 ?\) Explain. $$\begin{array}{|c|c|c|} \hline t=a & t=b & \text { Average speed }=\frac{d(b)-d(a)}{b-a} \\ \hline 3 & 3.5 & \\ 3 & 3.1 & \\ 3 & 3.01 & \\ 3 & 3.001 & \\ 3 & 3.0001 & \\ \hline \end{array}$$

The relative value of currencies fluctuates every day. When this problem was written, one Canadian dollar was worth 0.9766 U.S. dollars. (a) Find a function \(f\) that gives the U.S. dollar value \(f(x)\) of x Canadian dollars. (b) Find \(f^{-1}\). What does \(f^{-1}\) represent? (c) How much Canadian money would \(\$ 12,250\) in U.S. currency be worth?

In Example 7 and Exercises 82 and 83 we are given functions whose graphs consist of horizontal line segments. Such functions are often called step functions, because their graphs look like stairs. Give some other examples of step functions that arise in everyday life.
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