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

Compute the difference quotient $\frac{f(x+h)-f(x)}{h}, \quad h \neq 0$ Whenever possible, simplify the expression so that the resulting expression is defined when $h=0$.$f(x)=\sqrt{x+3}$ Hint refer to Example 19 Section 5.

$$\frac{1}{\sqrt{x+h+3}+\sqrt{x+3}}$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Oregon State University

McMaster University

Idaho State University

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

03:57

Compute the difference quo…

02:20

01:39

03:33

01:42

02:15

02:32

02:42

Find the difference quotie…

01:47

01:26

01:58

for this problem. We've been given a function f of X equals the square root of X plus three and we want to compute the difference quotient, which is f of X plus h minus ffx over H. Now, if you can see from our numerator, we have at the F function appearing twice once we're going to evaluate it at X plus h and the second time we're going to evaluate it at just X. And just as a side note, we're told that at each does not equal zero. So I'm not dividing by zero. In this case, however, we're gonna try to evaluate this and get it in such a way that I don't have that zero denominator anymore. So that's gonna be one of our goals when we go to simplify this. But first, let's actually put our function into this difference quotient. First f of X plus h. That's the square root of X plus H plus three, and I'm subtracting f of X, which is the square root of X plus three and it's all over H now where do we go from here to evaluate this or to simplify it? I still have that h in the denominator and I want to get rid of it now in your problem here it has a hint and it's referring you back to example 19 from the book. What that tells us is we can use. Remember we talked about fractions back? Oh, probably Maybe Algebra two had a radical in the denominator. We would always rationalize the denominator by multiplying by that square root of two top and bottom. If what we had was something that just maybe had two terms one plus the square root of two. We can't just multiply that by square root of two. So what we would dio is we would use the conjugate same terms. But we change that innermost sign. And what that does is it rationalizes our denominator. Well, sometimes. And this problem is one of them. We're going to rationalize in reverse. Instead of rationalizing the denominator, we're going to rationalize the numerator. So in this case, I'm going to multiply top and bottom by the conjugate number, those same two terms. But I'm going to change that middle sign because it was a minus. I'm now going to make it plus and I do that top and bottom exactly the same. So let's see what we get when we multiply this well on the top, the first term squared squaring a radical is just going to give me what's under the radical. When I foil this, the outer in the inner will cancel that za whole point of picking the cons. You get the way we dio and then the last one's I'll be subtracting what's under the radical in that second piece, X plus three. And my denominator is just going to be hte times square root of H plus X plus three plus the square root of X plus three. Okay, The nice thing is, much of the numerator cancels. I have X minus X three, minus three. And what I have left is just a H on top. Well, that cancels with my H factor in the denominator. So all I have in the numerator is one. Am I Denominator will be the some of these two radicals

View More Answers From This Book

Find Another Textbook

Numerade Educator

04:32

In each of the following exercises, solve the given inequality.$$\frac{(…

03:14

Plot the lines found in Exercise $14$.

02:51

Determine (a) $f(x)$ ) and the domain of the composite function, (b) $g(f(x)…

02:01

Find the equations of two lines parallel to $x=2$, and 6 units from it.

04:13

Table 6 shows the median incomes, in dollars, of men and women in the United…

03:29

A rock is thrown down from the ledge of a mountain 200 feet above the ground…

05:25

Determine (a) $f(x)+g(x),$ (b) $f(x)-g(x),$ (c) $f(x) g(x)$ (d) $f(x) / g(x)…

01:23

In each of the following, solve the given quadratic equation exactly using t…

00:55

Determine the horizontal asymptotes, if they exists.$$f(x)=\frac{2}{x-5}…

01:12