Paul's Online Notes
Paul's Online Notes
Home / Algebra / Graphing and Functions / The Definition of a Function
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 3.4 : The Definition of a Function

13. The difference quotient for the function \(f\left( x \right)\) is defined to be,

\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]

Compute the difference quotient for the function \(f\left( x \right) = 2{x^2} - x\).

Show All Steps Hide All Steps

Start Solution

We’ll work this problem in parts. First let’s compute \(f\left( {x + h} \right)\).

\[f\left( {x + h} \right) = 2{\left( {x + h} \right)^2} - \left( {x + h} \right) = 2\left( {{x^2} + 2xh + {h^2}} \right) - \left( {x + h} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{2{x^2} + 4xh + 2{h^2} - x - h}}\] Show Step 2

Now we’ll compute \(f\left( {x + h} \right) - f\left( x \right)\) and do a little simplification.

\[\begin{align*}f\left( {x + h} \right) - f\left( x \right) & = 2{x^2} + 4xh + 2{h^2} - x - h - \left( {2{x^2} - x} \right)\\ & = 2{x^2} + 4xh + 2{h^2} - x - h - 2{x^2} + x = \require{bbox} \bbox[2pt,border:1px solid black]{{4xh + 2{h^2} - h}}\end{align*}\]

Be careful with the parenthesis when subtracting \(f\left( x \right)\). We need to subtract the function and so we need parenthesis around the whole thing to make sure we do subtract the function.

Show Step 3

We can now finish the problem by computing the full difference quotient.

\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h} = \frac{{4xh + 2{h^2} - h}}{h} = \frac{{h\left( {4x + 2h - 1} \right)}}{h} = \require{bbox} \bbox[2pt,border:1px solid black]{{4x + 2h - 1}}\]