audio read-through Prerequisites: Function Review; Difference Quotients

Precalculus requires a thorough understanding of functions and the language used to work with functions. The lessons below will give you the required review.

Be sure to click-click-click through some of the exercises in each of these lessons! The lessons will open in a new tab/window.

Then, do the lesson on this page to practice with difference quotients, which are particularly important in calculus.

Example: Difference Quotients

Consider a point $\,\bigl(x,f(x)\bigr)\,$ on the graph of a function $\,f\,.$

If $\,h\,$ is a small positive number, then $\,x+h\,$ lies a little to the right of $\,x\,.$

If $\,h\,$ is a small negative number, then $\,x+h\,$ lies a little to the left of $\,x\,.$

In both cases, $\,\bigl(x+h,f(x+h)\bigr)\,$ is a point on the graph of $\,f\,,$ likely close to the original point $\,(x,f(x))\,.$

The slope of the line through $\,\bigl(x,f(x)\bigr)\,$ and $\,\bigl(x+h,f(x+h)\bigr)\,$ is:

$$ \begin{align} \cssId{s13}{\text{slope}}\ &\cssId{s14}{= \frac{\text{change in $y$}}{\text{change in $x$}}}\cr\cr &\cssId{s15}{= \frac{f(x+h) - f(x)}{(x+h)-x}}\cr\cr &\cssId{s16}{= \frac{f(x+h)-f(x)}{h}} \end{align} $$

To slide the point $\,\bigl(x+h,f(x+h)\bigr)\,$
closer to
the original point $\,\bigl(x,f(x)\bigr)\,,$

To keep the graph uncluttered,
the labels for the new (closer) points
are not shown.

(Refresh the page to start over.)

Recall that the result of a division problem, $\,\frac{a}{b}\,,$ is called a quotient, and the result of a subtraction problem, $\,a - b\,,$ is called a difference.

The expression $$\frac{f(x+h)-f(x)}{h}$$ is called a difference quotient, because it is a quotient of differences:

In calculus, you often have to simplify difference quotients, trying to rename in a way that eliminates the $\,h\,$ in the denominator.

For example, if $\,f(x) = x^2 + 2x + 1\,,$ then:

$$ \begin{align} &\cssId{s30}{\frac{f(x+h) - f(x)}h}\cr &\quad \cssId{s31}{= \frac{\overbrace{(x+h)^2 + 2(x+h) + 1}^{f(x+h)} - (\overbrace{\vphantom{(x+h)^2} x^2 + 2x + 1}^{f(x)})}h}\cr &\qquad\ \ \cssId{s32}{\text{(function evaluation)}\strut} \cr\cr &\quad \cssId{s33}{= \frac{x^2 + 2xh + h^2 + 2x + 2h + 1 - x^2 - 2x - 1}{h}}\cr &\qquad\ \ \cssId{s34}{\text{(FOIL; distributive law)}\strut}\cr\cr &\quad \cssId{s35}{= \frac{2xh + h^2 + 2h}{h}}\cr &\qquad\ \ \cssId{s36}{(x^2 - x^2 = 0;\ \ 2x - 2x = 0;\ \ 1 - 1 = 0)\strut}\cr\cr &\quad \cssId{s37}{= \frac{h(2x + h + 2)}{h}} \cssId{s38}{= \frac hh\cdot (2x + h + 2)}\cr &\qquad\ \ \cssId{s39}{\text{(factor out the $h$)}\strut}\cr\cr &\quad \cssId{s40}{= 2x + h + 2}\cr &\qquad\ \ \cssId{s41}{\strut(\text{cancel: } \textstyle\frac{h}{h} = 1)}\cr \end{align} $$

Note that the final form, $\,2x + h + 2\,,$ has no $\,h\,$ in the denominator.

Note also that as $\,h\,$ gets closer and closer to $\,0\,,$ the expression $\,2x + h + 2\,$ gets closer and closer to $\,2x + 2\,.$

You'll learn in calculus that $\,2x+2\,$ is called the ‘derivative of $f\,$ and is denoted by $\,f'(x)\,$ (read as ‘$f$ prime of $\,x\,$’).

This new function $\ f'(x) = 2x + 2\ $ gives the slope of the tangent line to the graph of $\,f\,$ at the point $\,\bigl(x,f(x)\bigr)\,$!

Since $\,x\,$ can represent any real number, you might prefer to say:

This new function $\ f'(x) = 2x + 2\ $ gives the slopes of the tangent lines to the graph of $\,f\,$ at the points $\,\bigl(x,f(x)\bigr)\,$!

NOTE: The notation ‘$\,\Delta x\,$’ (read aloud as ‘delta $x$’) is often used instead of $\,h\,$ to denote a small change in $\,x\,.$ With this notation, the slope of the line through the point $\,\bigl(x,f(x)\bigr)\,$ and nearby point $\,\bigl(x+\Delta x,f(x+\Delta x)\bigr)\,$ is given by the difference quotient:

$$\cssId{s57}{\frac{f(x+\Delta x) - f(x)}{\Delta x}}$$

Concept Practice