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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 $h=0$.$f(x)=\sqrt{2 x-1}$ Hint: refer to Example 19 Section 5.

$$\frac{2}{\sqrt{2 x+2 h-1}+\sqrt{2 x-1}}$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Missouri State University

Campbell University

Harvey Mudd College

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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Find the difference quotie…

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for this problem. We've been given a function f of X equals the square root of two X minus one. And we want to use this function to complete to compute the difference quotient, which is f of X plus h minus f of X over H. Now we're told that H does not equal zero, so we don't have to worry that we're dividing by zero here. However, we would like to have our final result be in the form. Uh, that is defined herbal. If h does equal zero, which means we're gonna want to find some way to get that eight out of the denominator. So that's me something we'll look at down the line in the middle toward the end of this problem. For right now, let's get this function f into our difference. Um quotient. Now, the numerator we're going toe reference our function twice the first time we're gonna plug in X plus h. So that gives us the square root of two X plus H minus one minus our function just a X, which is two X minus one, and that's going to be over H. So I'm going to just this two times, X plus h I'm going to erase this and just rewrite it without the parentheses. Two X plus two h. Okay. How do we proceed from here? Your book has a hit. It refers you back to example 19 from section five of this of this unit. What that hit tells us is we're gonna want to use the conjugate now, if you remember what the continent is. We you probably saw this when you were doing radicals. You might have something like this 1/1, plus the square root of two. We always wanted to rationalize the denominator so we would multiply top and bottom, have to keep our fraction balanced by the conjugate. It's the same two terms, but we change that sign since it was plus, it becomes a minus and we do that top and bottom. And that removes the radical from the denominator. In this case, we're going to use the continent is well, but we're gonna be rationalizing the numerator in this case. So we're going to multiply top and bottom by the same number, and it's going to be these exact same two terms. What's going to change is that middle sign. I was subtracting. So now I'm going to add and I'm going to do that top and bottom. Okay, So what do we get when we multiply these? Well, the point of picking the conjugate the way we did was removed things. We foil it first, outer inner last, the outer and the inner are gonna be exactly the same, but with opposite signs that they're going to cancel. So over here, multiplying the firsts, multiplying a radical times itself that the square roots that's just going to give me what's under the radical minus will outer and inner cancel and was going to subtract the last term again. I'm squaring square roots, so I just have what's under the radical. And, yes, this is going to be a nice, long, complicated denominator, but it's just going to sit here. I'm not going to do anything else with it right now. So what do we have for the numerator? Ah, lot of things Cancel. I have a two x minus two x. I have a negative one minus and negative one. All I have left in the numerator is A to H, and even that could be simplified because I haven't h in the denominator that can cancel with the H in the numerator. All that's left in the numerator, is it to? And my denominator is two x plus two H minus one under that radical, plus the square root of two X minus one. In the process of this, we've gotten rid of that age and the denominator, which is something we have talked about wanting to do earlier. And what I'm left with is this fraction.

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