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Chapter 4: Problem 92

Answer each of the following. For a one-to-one function \(f,\) find \(\left(f^{-1} \circ f\right)(2),\) where \(f(2)=3\).

Short Answer

Expert verified
The value is 2.

Step by step solution

01

Identify the Inverse Function Property

For a one-to-one function, the inverse function property states that \(f^{-1}(f(x)) = x\) for any x in the domain of f.
02

Apply the Inverse Function Property

Given the function f(2) = 3, we need to find \( (f^{-1} \circ f)(2) \). Using the inverse function property, \( f^{-1}(f(2)) = 2 \).
03

Substitute the Known Value

We know that \( f(2) = 3 \), so \( f^{-1}(f(2)) = f^{-1}(3) \). However, since \( f^{-1}(f(x)) = x \), it simplifies directly to \( 2 \).

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Key Concepts

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

One-to-One Functions
Let's dive into the concept of one-to-one functions. A function is considered one-to-one (or injective) if different inputs produce different outputs. In simple terms, if you plug in two different values, you get two different results. This unique relationship makes it possible to define an inverse function. For instance, if we have a function \( f(x) \) such that \( f(a) = b \) and \( f(c) = d \), then if \( a eq c \), it's guaranteed that \( b eq d \) too. This characteristic is essential when considering inverse functions.
Function Composition
Understanding function composition can greatly help in grasping inverse functions. Function composition involves applying one function to the result of another function. Mathematically, if you have two functions \( f \) and \( g \), the composition is written as \( (g \, \text{circ} \, f)(x) \), which means you first apply \( f \) to \( x \) and then apply \( g \) to the result. In our specific exercise, we deal with \( (f^{-1} \, \text{circ} \, f)(2) \). This means we apply \( f \) to \( 2 \) first and then \( f^{-1} \) to the result. Given \( f(2)=3 \), it follows that \( f^{-1}(3)=2 \), thus resulting in 2.
Function Properties
Several properties of functions aid in understanding and solving such exercises. One key property of inverse functions for any function \( f \) is that \( f^{-1}(f(x)) = x \). This property is crucial when dealing with one-to-one functions. Knowing that \( f \) maps to its inverse smoothly assures us that \( f \) and \( f^{-1} \) are perfectly linked. Hence, when you compose a function with its inverse, it undoes any action the function performed, bringing you back to your initial input. This is why in our exercise, we see that \( (f^{-1} \, \text{circ} \, f)(2) \) returns \( 2 \).

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Most popular questions from this chapter

(Modeling) Solve each problem. See Example 11 . Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table. $$\begin{array}{c|c||c|c} \hline \text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\ \hline 0 & 1013 & 6000 & 472 \\ \hline 1000 & 899 & 7000 & 411 \\ \hline 2000 & 795 & 8000 & 357 \\ \hline 3000 & 701 & 9000 & 308 \\ \hline 4000 & 617 & 10,000 & 265 \\ \hline 5000 & 541 & & \\ \hline \end{array}$$ (a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure \(P\) at altitude \(x\). (b) Would a linear or an exponential function fit the data better? (c) The function $$ P(x)=1013 e^{-0.0001341 x} $$ approximates the data. Use a graphing calculator to graph \(P\) and the data on the same coordinate axes. (d) Use \(P\) to predict the pressures at \(1500 \mathrm{m}\) and \(11,000 \mathrm{m},\) and compare them to the actual values of 846 millibars and 227 millibars, respectively.

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Consumer Price Index The U.S. Consumer Price Index for the years \(1990-2009\) is approximated by $$ A(t)=100 e^{0.026 t} $$ where \(t\) represents the number of years after \(1990 .\) (since \(A(16)\) is about \(152,\) the amount of goods that could be purchased for \(\$ 100\) in 1990 cost about \(\$ 152\) in 2006 .) Use the function to determine the year in which costs will be \(100 \%\) higher than in \(1990 .\)
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