Paul's Online Notes
Home / Calculus I / Review / Functions
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 1-1 : Review : Functions

8. The difference quotient of a 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 $$\displaystyle y\left( z \right) = \frac{1}{{z + 2}}$$.

Show All Steps Hide All Steps

Hint : Don’t get excited about the fact that the function is now named $$y\left( z \right)$$, the difference quotient still works in the same manner it just has $$y$$’s and $$z$$’s instead of $$f$$’s and $$x$$’s now. So, compute $$y\left( {z + h} \right)$$, then compute the numerator and finally compute the difference quotient.
Start Solution
$y\left( {z + h} \right) = \frac{1}{{z + h + 2}}$ Show Step 2
$y\left( {z + h} \right) - y\left( z \right) = \frac{1}{{z + h + 2}} - \frac{1}{{z + 2}} = \frac{{z + 2 - \left( {z + h + 2} \right)}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}} = \frac{{ - h}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}}$

Note that, when dealing with difference quotients, it will almost always be advisable to combine rational expressions into a single term in preparation of the next step.

Show Step 3
$\frac{{y\left( {z + h} \right) - y\left( z \right)}}{h} = \frac{1}{h}\left( {h\left( {z + h} \right) - h\left( z \right)} \right) = \frac{1}{h}\left( {\frac{{ - h}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}}} \right) = \frac{{ - 1}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}}$

In this step we rewrote the difference quotient a little to make the numerator a little easier to deal with. All that we’re doing here is using the fact that,

$\frac{a}{b} = \left( a \right)\left( {\frac{1}{b}} \right) = \left( {\frac{1}{b}} \right)\left( a \right)$