/*! 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 100 What does it mean if a function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 100

What does it mean if a function \(f\) is increasing on an interval?

Short Answer

Expert verified
A function \(f\) is said to be increasing on an interval if for all pairs of numbers \(x\) and \(y\) in that interval, if \(x < y\) then \(f(x) < f(y)\). This means that as we move from left to right, the function values increase.

Step by step solution

01

Definition of an Increasing Function

A function \(f\) is said to be increasing on an interval if, for all pairs of numbers \(x\) and \(y\) in the interval, if \(x < y\) then \(f(x) < f(y)\). This means that as we move from left to right (i.e., as the \(x\) values increase), the function values, or \(f(x)\) values, also increase.
02

Demonstration with an Example

Consider a simple function like \( f(x) = x^2 \) on the interval \([0, +\infty)\). It can be seen that for every two numbers \(x\) and \(y\) in \([0, +\infty)\) such that \(x < y\), the values of \(f(x)\) will be less than the values of \(f(y)\). Hence, the function \(f\) is increasing on the interval \([0, +\infty)\).
03

Visual Representation

When you plot the function on a graph, an increasing function should always appear to rise from left to right. In the graph of \(f(x) = x^2\) for \([0,+\infty)\), the graph moves upward as we move from left to right.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. $$ -x^{2}-2 x+1=0 $$

Explain how to find the difference quotient of a function \(f\) \(\frac{f(x+h)-f(x)}{h},\) if an equation for \(f\) is given.

determine whether each statement makes sense or does not make sense, and explain your reasoning. To avoid sign errors when finding \(h\) and \(k,\) I place parentheses around the numbers that follow the subtraction signs in a circle's equation.

The function $$ f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95 $$ models the number of annual physician visits, \(f(x),\) by a person of age \(x .\) Graph the function in a \([0,100,5]\) by \([0,40,2]\) viewing rectangle. What does the shape of the graph indicate about the relationship between one's age and the number of annual physician visits? Use the \([\mathrm{TABLE}]\) or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} x^{2}+y^{2} &=9 \\ x-y &=3 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.