Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

In Exercises $29-44,$ find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for each function.$$f(x)=\sqrt{x^{2}+x+1}$$

$\frac{f(x+h)-f(x)}{h}=\frac{2 x+h+1}{\sqrt{(x+h)^{2}+(x+h)+1}+\sqrt{x^{2}+x+1}}$

Precalculus

Algebra

Chapter 1

Functions and Their Graphs

Section 2

Graphs of Functions

Algebra Topics That are Reviewed at the Start of the Semester

Functions

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

03:18

10:07

In Exercises $29-44,$ find…

01:35

02:07

01:27

01:25

03:28

01:32

03:57

01:54

01:34

01:39

01:19

02:06

02:15

In Exercises 37-44, find t…

03:38

02:05

In Exercises $21-45,$ find…

01:48

02:45

we have. The function in for fix is equal to X squared plus X. Blessed one under the square root. And we have to find the difference caution for this function Which is given by a 4th express hedge minus of F. Of X. All over the edge. So let's see how to find this one. So we have to first find 1/4 express it. Which means we have to replace all X and F. Forfeits by F. Express it. So therefore it will become express it all squared. We replaced the excellent exc wired as explicit and this explosives will uh be replaced as expressive the first one and I just square root -4 fixed. So we put therefore access cities. So therefore this will become x squared. Press express man under this coil it and all over the edge. So now we have this expression we can go hold and further simplify this by multiplying the numerator and denominator by the conjugate off the uh this expression if you consider this expression as A. And this is B. We have a form like a minus B. So the conjugate of this expression is going to be A plus B. So let's multiply by the cards you get. Uh so it is like this so we ride out this expression asset is that is uh explicit square plus express, hedge the first one under the root. So we wrote on the auto this system and then we have to change this sign to positive that is pressed and we write on the V. As it is the veto as it is. So therefore this will become X squared. Press express it's X squared plus X plus one under the square room and do it by the same condo ghetto. So we write it down once again. So therefore this is going to be explicit square plus express it first one and the square root. And I don't hear. Plus another term is X squared Plus X-plus one with a square road. So this is the denominator. Now let's see how to simplify this expression. So we have a minus B. Times of a press be So remember that we have this identity. Different sub squares A minus B, A minus B times off A plus B. This is equal to a square minus this quick. So therefore we just start to Squire this time that if you uh look at this identity, we have a scrotum which means basically we just have to acquire this first radical top. So if it's quite a discordant will go off. So we will get explicit all square plus express it Plus one which is quite the a. That is a square. And now we will square the b. So therefore the square root will go from from the extra from the bottom. So everything is negative. So therefore this will be negative. X squared uh minus x minus fun. All over hedge times this denominator we have this denominator factor here. So let's ride it. Also this will be explicit. All square. Press express it. First one under the square root plus X squared plus X plus one. Another spoiled whole bracket. Now let's simplify this. Uh numerator. So uh huh. For this we need to expand this explosive elsewhere. When we do that we will have X squared plus two X. Hedge plus X squared. Okay. Plus this explosive plus one. So you can can slow this X squared with this X squared. And we have no this one is This one is head squared. Yes. I don't hate square. So we can cancel this uh X squared and two X. Squared. And we have this past two X. And the next two weeks. This will get canceled. This pastor one and this 91 will get cancelled and finally will be left with right on the remaining terms. That is to exit plus head square plus this hatch. So these are the only three terms which are left out after. We simplify the numerator all over by the denominator terms were right on this denominator terms, explosives all squared plus explosive charge. Last one the square root plus X squared plus X plus one. This quiet room. What is that? Uh We have hedged term uh the numerator all the terms in the numerator has hatched terms. So we can basically cancel this this age. This will get cancelled with one time on power outage and this will become one. So therefore finally you can write on this as we have two X plus H plus one. So this is basically equal to X plus, hedge the first one over this. Uh This radical expression, we can put this radical expression so that is express. It's all square less expressive. The first one plus X squared plus X plus one. This is the denominator. So this is the final expression. That is a different question for the given function.

View More Answers From This Book

Find Another Textbook

Numerade Educator

02:35

In Exercises $53-76$, graph the piccewise-defined functions. State the domai…

00:51

In Exercises $45-52,$ find the average rate of change of the function from $…

01:58

Solve by factoring.$$y(y-5)^{3}-14(y-5)^{2}=0$$

01:11

Find the equation of a parabola with vertex at the origin, axis of symmetry …

01:46

The sales tax in Orange and Seminole Counties in Florida differs by only $0.…

01:22

In Exercises $23-34$, rewrite the quadratic function in standard form by com…

Evaluate the difference quotients using the same $f, F, G,$ and $g$ given fo…

In Exercises $29-44,$ find the difference quotient $\frac{f(x+h)-f(x)}{h}$ f…

00:46

In Exerciess $17-28,$ state the (a) domain, (b) range, and (c) $x$ -interral…

00:36

Evaluate the given quantities applying the following four functions:$f(x…