INTRODUCTION TO FUNCTION NOTATION
Go to my homepage for access to about 350 sequenced free web exercises like this one: Dr. Carol JVF Burns

A function is a rule that takes an input, does something to it,
and gives a unique corresponding output.

There is a special notation (called ‘function notation’) that is used to represent this situation:
if the function name is [beautiful math coming... please be patient] $\,f\,$, and the input name is [beautiful math coming... please be patient] $\,x\,$,
then the unique corresponding output is called [beautiful math coming... please be patient] $\,f(x)\,$.
The notation ‘ [beautiful math coming... please be patient] $f(x)\,$’ is read aloud as:   ‘ [beautiful math coming... please be patient] $f\quad$ of [beautiful math coming... please be patient] $\quad x\,$ ’.

What exactly is [beautiful math coming... please be patient]$\,f(x)\,$?
Answer: It is the output from the function [beautiful math coming... please be patient]$\,f\,$ when the input is [beautiful math coming... please be patient]$\,x\,\,$.

What exactly is [beautiful math coming... please be patient]$\,g(t)\,$?
Answer: It is the output from the function [beautiful math coming... please be patient]$\,g\,$ when the input is [beautiful math coming... please be patient]$\,t\,$.

Note that [beautiful math coming... please be patient]$\,f\,$ and [beautiful math coming... please be patient]$\,f(x)\,$ are different :
[beautiful math coming... please be patient]$f\,$ is the name of the function (the ‘rule’);
[beautiful math coming... please be patient]$f(x)\,$ is the output from this rule when the input is [beautiful math coming... please be patient]$\,x\,$.

It is often helpful to think of a function as a ‘box’.
You drop an input in the top, something happens to the input inside the box, and the output drops out the bottom.
The box is labeled with the name of the function.

The letter [beautiful math coming... please be patient] $\,f\,$ is commonly used as the name of a function, since it is the first letter in the word function.

If [beautiful math coming... please be patient] $\,\,x\,\,$ is dropped in the top of the box labeled [beautiful math coming... please be patient] $\,f\,$, then [beautiful math coming... please be patient] $\,f(x)\,$ comes out the bottom.
If [beautiful math coming... please be patient] $\,\,t\,\,$ is dropped in the top of the box labeled [beautiful math coming... please be patient] $\,g\,$, then [beautiful math coming... please be patient] $\,g(t)\,$ comes out the bottom.
If [beautiful math coming... please be patient] $\,x+2t\,$ is dropped in the top of the box labeled [beautiful math coming... please be patient] $\,h\,$,
then [beautiful math coming... please be patient] $\,h(x+2t)\,$ (read as ‘ [beautiful math coming... please be patient] $h$   of   [beautiful math coming... please be patient] $x+2t$ ’) comes out the bottom.

The equation [beautiful math coming... please be patient] $\quad f(x) = x + 2\quad$ is function notation that describes the following situation:
a function named [beautiful math coming... please be patient] $\,f\,$ acts on an input (here, indicated by [beautiful math coming... please be patient] $\,x\,$), and gives the output [beautiful math coming... please be patient] $\,f(x)\,$, which is equal to [beautiful math coming... please be patient] $\,x+2\,$.
Thus, [beautiful math coming... please be patient] $\quad f(x) = x + 2\quad$ describes the   ‘ add $2$ ’   function.

This same function [beautiful math coming... please be patient] $f$ could also be described by any of these:
[beautiful math coming... please be patient] $f(t) = t + 2$
[beautiful math coming... please be patient] $f(w) = w + 2$
[beautiful math coming... please be patient] $f(u) = u + 2$
The variable used locally to give a name to the input is called a dummy variable.
In the equation [beautiful math coming... please be patient] $\,f(x) = x + 2\,$, the dummy variable is [beautiful math coming... please be patient] $\,x\,$.
In the equation [beautiful math coming... please be patient] $\,f(t) = t + 2\,$, the dummy variable is [beautiful math coming... please be patient] $\,t\,$.

EXAMPLES:
Question: What does the function notation [beautiful math coming... please be patient] $\,g(7)\,$ represent?
Answer: the output from the function [beautiful math coming... please be patient] $\,g\,$ when the input is $\,7$
Question: Suppose [beautiful math coming... please be patient] $\,f(x) = x + 2\,$.   What is [beautiful math coming... please be patient] $\,f(3)\,$?
Answer: [beautiful math coming... please be patient] $\,f(3) = 3 + 2 = 5$
Question: Suppose [beautiful math coming... please be patient] $\,f(x) = x + 2\,$.   What is [beautiful math coming... please be patient] $\,f(x+5)\,$?
Answer: [beautiful math coming... please be patient] $\,f(x+5) = (x+5) + 2 = x + 7$
Question: Suppose [beautiful math coming... please be patient] $\,g(x) = 3x + 2\,$. Describe, in words, what the function [beautiful math coming... please be patient] $\,g\,$ does.
Answer: the function [beautiful math coming... please be patient] $\,g\,$ takes an input, multiplies by [beautiful math coming... please be patient] $\,3\,$, and then adds [beautiful math coming... please be patient] $\,2$
Question: Write function notation for the ‘multiply by [beautiful math coming... please be patient] $\,5\,$’ rule.
Use [beautiful math coming... please be patient] $\,f\,$ as the name of the function, and [beautiful math coming... please be patient] $\,t\,$ as the name for the input.
Answer: $\,f(t) = 5t$
Question: Write the function rule [beautiful math coming... please be patient] $\,f(x) = 5x - 2\,$ using the dummy variable [beautiful math coming... please be patient] $\,t\,$.
Answer: [beautiful math coming... please be patient] $\,f(t) = 5t - 2$

You can explore a ‘function box’ and practice function notation 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.)

Function Boxes and Function Notation

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

When you're done practicing, move on to:
More Practice with Function Notation

 
 
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(MAX is 19; there are 19 different problem types.)