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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 $h=0$.$f(x)=\frac{x+1}{x-3}$

$$\frac{-4}{(x-3)(x+h-3)}$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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).

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Compute the difference quo…

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Find and simplify the diff…

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Determine the difference q…

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Find the difference quotie…

for this problem. We've been given a function f of X equals X plus one divided by X minus three and we want to compute the difference quotient. Now, the difference question is f of X plus H minus F of X over H. And this difference quotes that this is something we're going to see, uh, throughout calculus. But for right now, let's evaluate it. My numerator. I'm gonna be going to my function twice The first time I'm evaluating it at X plus h and the second time, just a X. So the X plus h first, I'm gonna go to my function, and I'm gonna substitute in X plus H for X. So I have X plus a H plus one in the numerator X plus H minus three. In the denominator, I'm going to subtract ffx, which is X plus one and X minus three, and this whole thing is going to be over H Now we want to simplify this as much as we can because we're told that H does not equal zero. But we would like to evaluate this to the point where um it's defined when h zero and right now it's not some dividing by zero. So I wanna see if I can do something to manipulate this algebraic Lee in some way to get rid of that H and the denominator. So let's look at the numerator. I have two fractions. I can put those together with a common denominator. It's my common denominator is going to be X plus H minus three times X minus three. So my my numerator will be X plus H plus one times X minus three and I'm going to subtract X plus H minus three times X plus one. And the whole thing is over H so I can put that h in the denominator. Okay, this is going to be long, so I'm just gonna write it all out and we'll see what we get. First, let's multiply those first two factors. That gives me X squared minus three X plus H X minus three H plus X minus three. And now we're subtracting everything here is when I multiply these out, I'm gonna be changing the signs because I'm going to distribute that subtraction. So that's gonna be minus X squared minus X up. Sorry. Hit the wrong On their minus h X minus H. Now there's gonna be plus plus three X plus three. And all of that is over my denominator, which I'm not going to change at the moment. Okay, that's kind of long. Let's see what we have. Some things were gonna cancel. I have X squared minus x squared. I have X minus X. I have HX minus h x Negative three X plus three X and I have negative three plus three. So with all of that, the Onley things that don't cancel. I have a minus four h left on top and again haven't changed my denominators. That's X plus H minus three X minus three times h. Well, we have one more thing. We can cancel because we have a factor of H in the numerator and the denominator. So what that leaves me with is negative four over these two factors.

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