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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 $h=0$.$f(x)=x^{2}-5 x+3$

$$2 x+ h-5$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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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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Simplify the difference qu…

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for this problem. We've been given a function f of X equals X squared minus five X plus three. And using this function, we're going to compute the difference quotient f of X plus h minus f of X over h. So for this type of problem, what we're going to do is we're going to be taking our function of F and we're evaluating at two places f of X, which is pretty straightforward. That's just what we have in our function and then f of X plus h. Which means every time we have an X in our original function, we're gonna plug in or substitute and X plus h. So let's put this together first f of X plus h every X. We're going to substitute X plus a h. So I have X plus h squared minus five times X plus H plus three and we're going to subtract ffx. Make sure we put this in parentheses. Since we're subtracting the entire amount, we're going to need Thio distribute that subtraction sign and we're dividing by H and we can see from the problem. H is not equal to zero. We're not going to be dividing by zero here. So let's get rid of our parentheses and see what we have. X plus H squared. That's X squared plus two X h plus H squared. Hey minus five x minus five H plus three minus X squared plus five x minus three. So I've distributed all of my subtraction signs and that's over H. So let's see what we have that we can cancel. I have a plus three under minus three on top. I have a plus X squared in a minus X squared and a plus five x and a minus five x. So what I have on top is two X h plus H squared minus five H over H. Now every single term has an H in it, so I can cancel that out. That's going to leave me two x plus h minus five.

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