/*! 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 23 Let \(f_{1}(x)=x, f_{2}(x)=\frac... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 23

Let \(f_{1}(x)=x, f_{2}(x)=\frac{1}{x}, f_{3}(x)=1-x\), \(f_{4}(x)=\frac{x}{x-1}, f_{5}(x)=\frac{1}{1-x},\) and \(f_{6}(x)=\frac{x-1}{x}\). a) Determine the following. i) \(f_{2}\left(f_{3}(x)\right)\) ii) \(\left(f_{3} \circ f_{5}\right)(x)\) iii) \(f_{1}\left(f_{2}(x)\right)\) iv) \(f_{2}\left(f_{1}(x)\right)\) b) \(f_{6}^{-1}(x)\) is the same as which function listed in part a)?

Short Answer

Expert verified
i) \(\frac{1}{1-x}\) ii) \(\frac{x}{x-1}\) iii) \(\frac{1}{x}\) iv) \(\frac{1}{x}\) b) \(f_{6}^{-1}(x) = f_{4}(x)\).

Step by step solution

01

Find \(f_{2}(f_{3}(x))\)

First, find \(f_{3}(x)\): \[ f_{3}(x) = 1 - x \]Next, substitute \(f_{3}(x)\) into \(f_{2}\): \[ f_{2}(1 - x) = \frac{1}{1 - x} \]
02

Find \(f_{3} \circ f_{5}(x)\)

First, find \(f_{5}(x)\): \[ f_{5}(x) = \frac{1}{1 - x} \]Next, substitute \(f_{5}(x)\) into \(f_{3}\): \[ f_{3}\left(\frac{1}{1 - x}\right) = 1 - \frac{1}{1 - x} = \frac{(1 - x) - 1}{1 - x} = \frac{-x}{1 - x} = \frac{x}{x - 1} \]
03

Find \(f_{1}(f_{2}(x))\)

First, find \(f_{2}(x)\): \[ f_{2}(x) = \frac{1}{x} \]Next, substitute \(f_{2}(x)\) into \(f_{1}\): \[ f_{1}\left(\frac{1}{x}\right) = \frac{1}{x} \]
04

Find \(f_{2}(f_{1}(x))\)

First, find \(f_{1}(x)\): \[ f_{1}(x) = x \]Next, substitute \(f_{1}(x)\) into \(f_{2}\): \[ f_{2}(x) = \frac{1}{x} \]
05

Verify functions and find \(f_{6}^{-1}(x)\)

Start with \(f_{6}(x)\): \[ f_{6}(x) = \frac{x-1}{x} \]Set \(f_{6}(y) = x\) and solve for \(y\): \[ x = \frac{y-1}{y} \implies xy = y - 1 \implies y(x-1) = 1 \implies y = \frac{1}{x-1} \]Thus, \(f_{6}^{-1}(x) = \frac{1}{x-1}\), which is equivalent to \(f_{2}(x)\) with a shift.

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Key Concepts

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

Precalculus
Precalculus acts as the bridge between Algebra and Calculus. Understanding functions, inverses, and their compositions lie at the heart of this subject.
We often explore how functions behave, both individually and within combinations, to prepare for more complex calculus ideas.
In the given problem, we deal with six distinct functions, each with their unique forms and behaviors.
Especially in precalculus, it's crucial to grasp how these functions operate when composed together.
Inverse Functions
Inverse functions are a fundamental concept in mathematics. They reverse the operation of the original functions.
If we have a function \( f(x) \), its inverse function, denoted as \( f^{-1}(x) \), will satisfy: \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \).
In the exercise, part (b) involves finding \( f_6^{-1}(x) \). By solving, we discover it is equivalent to the function \( f_2 \), represented as \( \frac{1}{x-1} \). This reveals its operation as the inverse of the specified function.
Composite Functions
Composite functions involve the combination of two or more functions. The composition \( (f \circ g)(x) \) means we apply the function \( g \) first, then \( f \) to the result of \( g(x) \).
In the exercise, we see compositions such as \( f_2(f_3(x)) \) and \( f_3 \circ f_5(x) \).
The challenge is to substitute one function into the other correctly and simplify. For example, \( f_2(f_3(x)) = \frac{1}{1-x} \), while \( f_3(f_5(x)) = \frac{x}{x-1} \).
Understanding these compositions enriches our ability to handle multiple operations systematically.
Algebraic Manipulation
Algebraic manipulation is the skill of transforming expressions into a more useful or simplified form.
This may involve combining like terms, factoring, expanding, or simplifying fractions.
In the solution, algebraic manipulation helps find the compositions of functions and their inverses. For instance, deriving \( f_3 \circ f_5(x) \) required careful simplification: \( 1 - \frac{1}{1 - x} = \frac{x}{x-1} \).
Another example is determining \( f_6^{-1}(x) \) by solving and rearranging terms until reaching \( f_2(x) \).
Mastering these techniques is crucial in precalculus for dealing with complex equations effortlessly.

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Most popular questions from this chapter

In general, two functions \(f(x)\) and \(g(x)\) are inverses of each other if and only if \(f(g(x))=x\) and \(g(f(x))=x .\) Verify that the pairs of functions are inverses of each other. a) \(f(x)=5 x+10\) and \(g(x)=\frac{1}{5} x-2\) b) \(f(x)=\frac{x-1}{2}\) and \(g(x)=2 x+1\) c) \(f(x)=\sqrt[3]{x+1}\) and \(g(x)=x^{3}-1\) d) \(f(x)=5^{x}\) and \(g(x)=\log _{5} x\)

According to Einstein's special theory of relativity, the mass, \(m,\) of a particle moving at velocity \(v\) is given by \(m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\), where \(m_{0}\) is the particle's mass at rest and \(c\) is the velocity of light. Suppose that velocity, \(v,\) in miles per hour, is given as \(V=t^{3}\). a) Express the mass as a function of time. b) Determine the particle's mass at time \(t=\sqrt[3]{\frac{c}{2}}\) hours.

For each pair of functions, determine \(h(x)=f(x)-g(x)\) a) \(f(x)=6 x\) and \(g(x)=x-2\) b) \(f(x)=-3 x+7\) and \(g(x)=3 x^{2}+x-2\) c) \(f(x)=6-x\) and \(g(x)=(x+1)^{2}-7\) d) \(f(x)=\cos x\) and \(g(x)=4\)

A Ferris wheel rotates such that the angle, \(\theta,\) of rotation is given by \(\theta=\frac{\pi t}{15},\) where \(t\) is the time, in seconds. A rider's height, \(h,\) in metres, above the ground can be modelled by \(h(\theta)=20 \sin \theta+22\) a) Write the equation of the rider's height in terms of time. b) Graph \(h(\theta)\) and \(h(t)\) on separate sets of axes. Compare the periods of the graphs.

The daily costs for a hamburger vendor are \(\$ 135\) per day plus \(\$ 1.25\) per hamburger sold. He sells each burger for \(\$ 3.50,\) and the maximum number of hamburgers he can sell in a day is 300. a) Write equations to represent the total cost, \(C,\) and the total revenue, \(R,\) as functions of the number, \(n,\) of hamburgers sold. b) Graph \(C(n)\) and \(R(n)\) on the same set of axes. c) The break-even point is where \(C(n)=R(n) .\) Identify this point. d) Develop an algebraic and a graphical model for the profit function. e) What is the maximum daily profit the vendor can earn?
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