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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 $h=0$.$f(x)=x^{2}+3 x+9$

$$2 x+h+3$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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Harvey Mudd College

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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 the difference quotie…

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Find and simplify the diff…

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it's this problem. We have been given a function f of X equals X squared plus three x plus nine. And for this problem, we want to compute the difference quotient, which is f of X plus h minus f of X, divided by H. And this is, ah, question that we'll see often when it comes to calculus. So the numerator we're going to be, uh, having two different, uh, iterations of our function. F one evaluated Justin X and one evaluated at X plus h. So let's do that X plus h one first. Everywhere in my function f where I now have an ex, I'm going to substitute in X plus h so that gives me X Plus h squared plus three times X plus H Plus nine. And I'm going to subtract from that f of X, which is X squared plus three x plus nine. Now you are subtracting the entire things that don't forget to put parentheses around that ffx because we have to distribute that subtraction sign and everything is over. H Okay, let's expand this out. Let's get rid of our parentheses, distribute the subtraction sign and see what we have first X Plus h squared that is X squared plus two X h plus h squared plus three x plus three h plus nine. And now we subtract all three terms minus X squared minus three X minus nine. Poop over H but what do we have? I have a plus three x and uh minus three X. They cancel plus nine minus nine Positive X squared Negative X squared. The terms I have left on top are two x h plus h squared plus three h over H. Well, every single term in my numerator has an h, so I can cancel that with the H on the bottom, and that gives me two x plus h plus three as the value of my difference quotient.

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