/*! 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 41 Show that the sum of two increas... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 41

Show that the sum of two increasing functions is increasing.

Short Answer

Expert verified
Given two increasing functions f(x) and g(x), we want to show that their sum h(x) = f(x) + g(x) is also increasing. Based on the definition of increasing functions, for any x1 < x2, we have f(x1) < f(x2) and g(x1) < g(x2). Analyzing h(x) at points x1 and x2, we get h(x1) = f(x1) + g(x1) and h(x2) = f(x2) + g(x2). Adding the inequalities, we obtain h(x1) < h(x2) for any x1 < x2, proving that h(x) is indeed an increasing function.

Step by step solution

01

Define the functions and their properties

Let f(x) and g(x) be two increasing functions. By the definition of increasing functions, for any x1 < x2, we have: 1. f(x1) < f(x2) 2. g(x1) < g(x2) Now, let h(x) = f(x) + g(x) be the sum of these two increasing functions. We need to prove that h(x) is increasing, which means for any x1 < x2, h(x1) < h(x2).
02

Analyze the sum function h(x)

Let's analyze the function h(x) at the points x1 and x2 where x1 < x2. Since h(x) is the sum of f(x) and g(x), we can write h(x1) and h(x2) as: 1. h(x1) = f(x1) + g(x1) 2. h(x2) = f(x2) + g(x2) Now, to prove that h(x) is increasing, we need to show that h(x1) < h(x2).
03

Apply the properties of increasing functions

We know that both f(x) and g(x) are increasing functions; therefore, f(x1) < f(x2) and g(x1) < g(x2). Let's add these two inequalities: f(x1) + g(x1) < f(x2) + g(x2) Notice that the left-hand side is equal to h(x1) and the right-hand side is equal to h(x2). Thus, we get: h(x1) < h(x2)
04

Conclude the proof

We have shown that for any x1 < x2, h(x1) < h(x2). This means that h(x) is an increasing function. Therefore, we have proven that the sum of two increasing functions is also an increasing function.

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Most popular questions from this chapter

Suppose you are exchanging cur. rency in the London airport. The currency exchange service there only makes transactions in which one of the two currencies is British pounds, but you want to exchange dollars for Euros. Thus you first need to exchange dollars for British pounds, then exchange British pounds for Euros. At the time you want to make the exchange, the function \(f\) for exchanging dollars for British pounds is given by the formula \(f(d)=0.66 d-1\) and the function \(g\) for exchanging British pounds for Euros is given by the formula \(g(p)=1.23 p-2\) The subtraction of 1 or 2 in the number of British pounds or Euros that you receive is the fee charged by the currency exchange service for each transaction. Which process gives you more Euros: exchanging $$\$ 100$$ for Euros twice or exchanging $$\$ 200$$ for Euros once?

Draw the graph of a function that is decreasing on the interval [-2,1] and increasing on the interval [1,5] .

Suppose \(f\) is an even function and \(g\) is an odd function. Show that \(f \circ g\) is an even function.

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ What is the domain of \(g ?\)

Check your answer by evaluating the appropriate function at your answer. Suppose \(f(x)=3 x+2\). Find a formula for \(f^{-1}\).
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