Graphs of Functions

Recall that a function is a rule that takes an input, does something to it, and gives an output.
Each input has exactly one output.

If the function name is $\,f\,$, and the input name is $\,x\,$,
then the unique corresponding output is called $\,f(x)\,$ (which is read aloud as ‘$\,f\,$ of $\,x\,$’).

This use of the notation $\,f(x)\,$ to represent the unique output from the function $\,f\,$ when the input is $\,x\,$ is called function notation.

When you're working with a function,
it's critical that you understand the relationship between its inputs and their corresponding outputs.
That is, it's critical that you understand the function's (input,output) pairs.
Of course, there are usually infinitely many of these (input,output) pairs.

For example, consider the squaring function—the function that takes a real number input, and squares it.
When the input is $\,3\,$, the output is $\,3^2 = 9\,$.
Thus, $\,(3,9)\,$ is an (input,output) pair.

When the input is $\,4\,$, the output is $\,4^2 = 16\,$.
Thus, $\,(4,16)\,$ is an (input,output) pair.

When the input is $\,-3\,$, the output is $\,(-3)^2 = 9\,$.
Thus, $\,(-3,9)\,$ is an (input,output) pair.

Here's a table (at right) that summarizes a few of the infinitely-many (input,output) pairs. Of course, it's impossible to list them all.

When these points are plotted in an

$\,xy\,$-coordinate system (see below),
with the inputs along the $\,x\,$-axis and the outputs along the $\,y\,$-axis,
a shape clearly emerges in the coordinate plane.

graph of the squaring function

SOME (INPUT,OUTPUT) PAIRS
FOR THE SQUARING FUNCTION

input	output	(input,output)
$-3$	$9$	$(-3,9)$
$-2$	$4$	$(-2,4)$
$-1$	$1$	$(-1,1)$
$0$	$0$	$(0,0)$
$\frac12$	$\frac14$	$(\frac12,\frac14)$
$1$	$1$	$(1,1)$
$2.3$	$5.29$	$(2.3,5.29)$
$\pi$	$\pi^2$	$(\pi,\pi^2)$

The picture of all the points of the form $\,(x,x^2)\,$ is called the graph of the squaring function.

You can explore this graph using GeoGebra.
GeoGebra is a free, multi-platform, dynamic mathematics software program that joins geometry, algebra and calculus.
(Dr. Fisher pronounces ‘GeoGebra’ like ‘Algebra’ except with a ‘Geo’ at the beginning.)
Click on the link below and have fun! (Please be patient. It may take a few minutes for GeoGebra to load. The link opens in a new window.)

Explore the Squaring Function with GeoGebra

Now it's time to make things precise:

DEFINITION graph of a function

Let $\,f\,$ be a function with domain

$\,\text{dom}(f)\,$.
The graph of $\,f\,$ is the picture of all its (input,output) pairs.

Precisely:

$$ \text{graph of } f = \{(x,f(x))\ |\ x\in\text{dom}(f)\} $$ (Read this aloud as: “The graph of $\,f\,$ is the set of all points of the form $\,x\,$, comma, $\,f\,$ of $\,x\,$, with the property that $\,x\,$ is in the domain of $\,f\,$.”)

When you graph a function:
-- the inputs (the first coordinates of the points) are placed along the $\,x$-axis;
-- the outputs (the second coordinates of the points) are placed along the $\,y$-axis.

The graph itself should then be labeled $\,y=f(x)\,$;
this indicates that the $\,y$-value of each point is
the output from the function $\,f\,$ when the input is $\,x\,$.

Different names (other than $\,x\,$ and $\,y\,$) may certainly be used for the inputs and outputs;
the graph should be labeled accordingly.

The sketch at right illustrates all the key features of a graph.
The input (horizontal) axis is labeled as $\,x\,$.
The output (vertical) axis is labeled as $\,y\,$.
The graph itself is labeled as $\,y = f(x)\,$.
A couple specific (input,output) pairs are shown.

You should recognize this as the graph of the squaring function.
That is,

$\,f(x) = x^2\,$.
Thus, $\,f(-2) = (-2)^2 = 4\,$ and $\,f(1) = 1^2 = 1\,$.

Alternate names for inputs and outputs have been chosen for the graph at left.
The input (horizontal) axis is labeled as $\,t\,$.
The output (vertical) axis is labeled as $\,w\,$.
The graph itself is labeled as $\,w=g(t)\,$.
This says that a function named $\,g\,$ is acting on inputs named $\,t\,$
and producing outputs named $\,w\,$.
A couple specific (input,output) pairs are shown.

You may have guessed that this is the graph of the cubing function.
That is,

$\,g(t) = t^3\,$.
Thus, $\,g(-1) = (-1)^3 = -1\,$ and $\,g(2) = 2^3 = 8\,$.

You can use GeoGebra to play with graphs of functions to your heart's content. Have fun!
(Please be patient. It may take a few minutes for GeoGebra to load.)

Play with Functions using GeoGebra

Two Ways to Ask the Same Question

Here are two things you will frequently be asked to do:

Graph the function $\,f(x) = x^2\,$.
Or, more simply: Graph $\,f(x) = x^2\,$.
Or, slightly more correctly: Graph the function $\,f\,$ defined by $f(x ) = x^2\,$.

Here, you're being asked for a picture of all the (input,output) pairs for $\,f\,$.
That is, you're being asked for a picture of all the points of the form $\,(x,f(x)) = (x,x^2)\,$.
Graph the equation $\,y = x^2\,$.
Or, more simply: Graph $\,y = x^2\,$.

The sentence ‘$\,y= x^2\,$’ is an equation in two variables.
To graph this sentence means to show all the choices for $\,x\,$ and $\,y\,$ that make it true.
Thus, you are being asked to show a picture of all the points of the form $\,(x,y) = (x,x^2)\,$.

Two different-sounding questions, but exactly the same answer.
It's very important that you are comfortable with these interchangeable ways that you might be asked for a graph.

Example: Reading Information From a Graph

The graph of a function $\,f\,$ is shown at right.
Read the following information from the graph:

$f(1)$
$f(\frac12)$
$f(1+0.0001)$
$f(1)+f(0.0001)$

SOLUTIONS:

$f(1)=10$
A solid (filled-in) dot indicates that a point is actually there;
it indicates an (input,output) pair.
A hollow (empty; not filled-in) dot indicates that a point is not there.
$f(\frac12)=5$
$f(1+0.0001)=f(1.0001)=10$
$f(1)+f(0.0001)=10+5=15$

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Basic Models You Must Know

(MAX is 16; there are 16 different problem types.)