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Chapter 8: Problem 5

Given \(F(x)=2 x+1\) and \(G(x)=\frac{x-1}{x+2},\) find: a. \(F(G(1))\) d. \(F(F(0))\) b. \(G(F(-2))\) e. \((F \circ G)(x)\) c. \(F(G(2))\) f. \((G \circ F)(x)\)

Short Answer

Expert verified
a. 1b. 3c. 4d. 3e. \(\frac{3x}{x+2}\)f. \(\frac{2x}{2x+3}\)

Step by step solution

01

Identify the functions

We are given two functions: 1. \( F(x) = 2x + 1 \) 2. \( G(x) = \frac{x - 1}{x + 2} \)
02

Step 1a: Find F(G(1))

First, find \( G(1) \): \( G(1) = \frac{1 - 1}{1 + 2} = \frac{0}{3} = 0 \).Then, find \( F(0) \): \( F(0) = 2 \cdot 0 + 1 = 1 \).Thus, \( F(G(1)) = 1 \).
03

Step 1b: Find F(F(0))

First, find \( F(0) \): \( F(0) = 2 \cdot 0 + 1 = 1 \).Then, find \( F(1) \): \( F(1) = 2 \cdot 1 + 1 = 3 \).Thus, \( F(F(0)) = 3 \).
04

Step 1c: Find G(F(-2))

First, find \( F(-2) \): \( F(-2) = 2 \cdot (-2) + 1 = -4 + 1 = -3 \).Then, find \( G(-3) \): \( G(-3) = \frac{-3 - 1}{-3 + 2} = \frac{-4}{-1} = 4 \).Thus, \( G(F(-2)) = 4 \).
05

Step 1d: Find F(G(2))

First, find \( G(2) \): \( G(2) = \frac{2 - 1}{2 + 2} = \frac{1}{4} \).Then, find \( F\left( \frac{1}{4} \right) \): \( F\left( \frac{1}{4} \right) = 2 \cdot \frac{1}{4} + 1 = \frac{2}{4} + 1 = \frac{1}{2} + 1 = \frac{3}{2} \).Thus, \( F(G(2)) = \frac{3}{2} \).
06

Step 1e: Find (F \circ G)(x)

First, find \( G(x) \): \( G(x) = \frac{x - 1}{x + 2} \).Then, substitute \( G(x) \) into \( F(x) \): \( F(G(x)) = F\left( \frac{x - 1}{x + 2} \right) = 2 \left( \frac{x - 1}{x + 2} \right) + 1 = \frac{2(x - 1)}{x + 2} + 1 \).Combine the terms to get a common denominator: \( \frac{2(x - 1)}{x + 2} + 1 = \frac{2(x - 1)}{x + 2} + \frac{x + 2}{x + 2} = \frac{2x - 2 + x + 2}{x + 2} = \frac{3x}{x + 2} \).Thus, \( (F \circ G)(x) = \frac{3x}{x + 2} \).
07

Step 1f: Find (G \circ F)(x)

First, find \( F(x) \): \( F(x) = 2x + 1 \).Then, substitute \( F(x) \) into \( G(x) \): \( G(F(x)) = G(2x + 1) = \frac{(2x + 1) - 1}{(2x + 1) + 2} = \frac{2x}{2x + 3} \).Thus, \( (G \circ F)(x) = \frac{2x}{2x + 3} \).

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Key Concepts

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

Algebraic Functions
Algebraic functions are functions that involve mathematical operations such as addition, subtraction, multiplication, division, and raising to a power. They are built from basic algebraic expressions. For example, consider the functions given in the exercise: the linear function \(F(x) = 2x + 1\) and the rational function \(G(x) = \frac{x - 1}{x + 2}\).

Linear functions like \(F(x)\) represent straight lines if graphed, following the formula \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Rational functions like \(G(x)\) involve ratios of polynomial expressions. They are not linear because their graphs can exhibit asymptotic behavior.
Function Evaluation
Function evaluation is all about finding the output of a function for a given input. For example, to evaluate \(F(0)\) in the function \(F(x) = 2x + 1\), we simply substitute \(x\) with \(0\), resulting in \(F(0) = 2(0) + 1 = 1\).

The process is similar for more complex functions. For example, evaluating \(G(1)\) in \(G(x) = \frac{x - 1}{x + 2}\) would involve substituting \(x\) with \(1\), giving \(G(1) = \frac{1 - 1}{1 + 2} = \frac{0}{3} = 0\).

Function evaluation helps to understand how the input relates to the output in a function's operational context.
Composite Functions
Composite functions involve the combination of two functions to create a new function. This is done by taking the output of one function and using it as the input for another. The notation \( (F \text{ \textopenbullet } G)(x) \) represents the composite function \(F(G(x))\), meaning that \(G(x)\) is computed first and its result is used to evaluate \(F\).

For example, in the exercise, to find \(F(G(1))\), you need to first compute \(G(1)\) and then use the result as input for \(F\). Here, \(G(1) = 0\), so \(F(G(1)) = F(0) = 1\).

Understanding composite functions is essential for dealing with more complex mathematical operations and various application scenarios.
Rational Functions
Rational functions are specific types of functions represented by the ratio of two polynomials. They are written in the form \( R(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\).

An example given in the exercise is \( G(x) = \frac{x - 1}{x + 2} \). These functions often have vertical asymptotes where the denominator equals zero, making the function undefined at those points.

Rational functions are useful in various fields such as physics and economics, where they can model relationships involving rates and proportions. Understanding how to manipulate and evaluate them is crucial for solving a wide range of mathematical problems.

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Most popular questions from this chapter

Market research suggests that if a particular item is priced at \(x\) dollars, then the weekly profit \(P(x)\), in thousands of dollars, is given by the function $$ P(x)=-9+\frac{11}{2} x-\frac{1}{2} x^{2} $$ a. What price range would yield a profit for this item? b. Describe what happens to the profit as the price increases. Why is a quadratic function an appropriate model for profit as a function of price? c. What price would yield a maximum profit?

For each of the following quadratics with their respective vertices, calculate the distance from the vertex to the focal point. Then determine the coordinates of the focal point. a. \(f(x)=x^{2}-2 x-3\) with vertex at (1,-4) b. \(g(t)=2 t^{2}-16 t+24\) with vertex at (4,-8)

(Graphing program optional.) a. On separate grids sketch the graphs of \(f(x)=\sqrt{-x+2}\) and \(g(x)=-\sqrt{x+2}\) b. Using interval notation, describe the domains of \(f(x)\) and \(g(x)\) c. Using interval notation, describe the ranges of \(f(x)\) and \(g(x)\) d. What is the simplest function \(h(x)\) from which both \(f(x)\) and \(g(x)\) could be created? e. Describe the transformations of \(h(x)\) to obtain \(f(x)\). Of \(h(x)\) to obtain \(g(x)\) f. Does the graph of \(f(x), g(x),\) or \(h(x)\) have any symmetries (across the \(x\) - or \(y\) -axis, or about the origin)?

Let \(f(x)=m x+b\) a. Does \(f(x)\) always have an inverse? Explain. b. If \(f(x)\) has an inverse, find \(f^{-1}(x)\). c. Using the formula for \(f^{-1}(x)\), explain in words how, given any linear equation (under certain constraints), you can find the inverse function knowing the slope \(m\) and \(y\) -intercept \(b\).

Given the function \(g(t)\), identify the simplest function \(f(t)\) (linear, power, exponential, or logarithmic) from which \(g(t)\) could have been constructed. Describe the transformations that changed \(f(t)\) to \(g(t)\) a. \(g(t)=\frac{t-1}{2}\) b. \(g(t)=3\left(\frac{1}{2}\right)^{t+4}\) c. \(g(t)=\frac{-7}{t-5}-2\)
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