Section
3.12 Using Function Notation
The example of distance from Earth to Moon illustrates that function
notation gives us representational power. With function notation, we
can represent relationships between specific values of quantities
conceptually before we even know how their values might actually be related.
You will see later that using function notation also allows us to
investigate types of functions and their properties.
Conventions for Function Definitions Using Function Notation
Functions can be defined two ways: conceptually or computationally.
A function defined conceptually describes the nature of the
relationship between two variables. The example of distance between Earth
and Moon, given earlier, defined this distance conceptually in relation to
the passage of time.
A function defined computationally specifics how to calculate values
of the dependent variable from values of the independent variable.
Computational definitions of functions are the centerpiece of mathematical
modeling. Functions defined computationally allow us to gain precise
numerical information about the phenomena we model. At the same time, when
modeling any situation, a conceptual definition should precede a
computational definition.
It is imperative that you specify the domain of a function’s independent
variable, regardless of whether you define the function conceptually,
computationally, or both. In this textbook, we will assume that the real
numbers comprise any independent variable’s domain unless stated otherwise
or unless restrictions are required by context.
You have seen many examples of functions defined computationally. The
statement $y = u(13.76 ‑ 2u)(16.42 ‑ 2u)$, u
a real number, defines y as a function of u computationally.
However, this function has no name, so we have no way to speak of it aside
from the repeating the entire statement. We also have no way to represent
the value of y that corresponds to a specific value of u.
Function notation solves these problems. We would instead write
$V(u)=u(13.762u)(16.422u)$
In this statement, V is the name of the relationship between
independent and dependent variables, u represents values of the
independent variable, $V(u)$ represents the value of the dependent value
that corresponds to a value of u, and the expression
$u(13.76 ‑ 2u)(16.42 ‑ 2u)$ tells us how to calculate
$V(u)$ from a value of u. Figure 3.12.1 summarizes all this
graphically.
Figure 3.12.1. Components of a function defined computationally using
function notation
We refer to a function by stating its name. Thus, V is the
function defined in Figure 3.12.1. It is a mistake to say that $V(u)$ is
the function defined in Figure 3.12.1; $V(u)$ represents the value of the
dependent variable that corresponds to a value of u. The
function is V, not $V(u)$.
You should practice forming an image of how a function defined
computationally (as in Figure 3.12.1 above) is related to the conceptual
definition of functions given in Section
3.11. The animation below might help.
How Function Notation Works
Prior sections stressed the importance that you understand and utilize the
representation power of function notation. It is equally important to have
an image of how function notation turns values of an independent variable
into values of a dependent variable. Figure 3.12.2 starts with a definition
of a function g in terms of an independent variable u. According to the
convention shown in Figure 3.12.1, the definition in Figure 3.12.2 names the
function (g), names the independent variable (u), and states
what to do with a value of u to produce the value of the dependent
variable that is associated with that value of u, represented by
$g(u)$.
Figure 3.12.2.a Evaluating g(3x) at x = 2.
Many students object to the expression $g(3x)$ in Figure 3.12.2a because the
definition of g uses the letter u, not the letter x.
However, it is important to understand that the letter u in Figure
3.12.2a represents a value of the independent variable. The
animation in Figure 3.12.2a shows that there is no conflict between using x
in one expression and u in the definition of g.
Once we assign a value to x, the statement $g(3x)$ invokes the
definition of g and passes the value of 3x to it.
When the number 2 is assigned as a value of x, 3x is 6, and
the statement $g(3x)$ invokes the definition of g with the number 6
as the value of u.
Figure 3.12.2b. Substituting “3x” for u in the definition of g.
The animation in Figure 3.12.2b shows a more formal way of thinking about
$g(3x)$. It invokes the definition of g with the expression 3x
as the value of u. If you substitute $x = 2$ after
substituting 3x for u, you will get g(6), just as in
Figure 3.12.2a.
Figure 3.12.3 displays the definition of the function u using an
independent variable u. This might seem confusing, but it is not.
The function’s name is u. Any letter can be used to represent values
of its independent variable. The animation in Figure 3.12.3 shows how the
different roles of function name and independent variable name removes any
possibility of confusion.
Figure 3.12.3. Conventions of function notation make letters’ roles
unambiguous.
Function Notation in GC
Ron Avitzur, the creator of GC, faced a problem caused by GC’s flexibility
in graphing equations and functions. A person can use letters like
f and g as parameters and they can use them as function names.
So someone could write an expression like $f(x + 5)$ to mean the
parameter f times the sum $(x + 5)$, or she could mean a
function named f evaluated at the value of the expression
$x + 5$.
Avitzur needed a way that people could signal to GC which way they intended
that GC interpret potentially ambiguous statements. He settled on the
convention that instead of typing f and shift9 to define a function using
function notation, you will type f and ctrl9. Typing ctrl9 causes GC to
produce parentheses, but with the meaning that they contain a letter or an
expression that gives a value to the function.
f shift9 x + 5 
→ 
f (x+5) 
Parameter f times the value of x+5

f ctrl9 x + 5 
→ 
f(x+5) 
Function f evaluated with the value of x+5 
A visual cue to tell function notation and parameter multiplication
apart is that the parentheses are closer to a function name than to a
parameter name.
GC does not require ctrl9 for builtin functions like sin or cos.
In fact, you will confuse GC if you use ctrl9 with a builtin function. Use
ctrl9 only with functions that you define. Do not use ctrl9 with
functions that are built into GC.
Multiple Independent Variables
GC allows functions defined in terms of several independent variables. For
example, the function S, below, converts a time given in hours,
minutes, and seconds into a number of seconds.
$$S(h,m,s) = 3600h+60m+s$$
To convert 23:40:17 into a number of seconds, enter $S(23,40,17)$. GC will
display 85217, which is the number of seconds in 23 hours, 40 minutes, and
17 seconds.
Exercise Set 3.12
Exercises 15 will give you practice defining functions in GC. Type
ctrlEnter to make a new line. For each function, describe what GC does to
display a graph as the value of x varies through its visible domain.

Define this function 
Type this 
You should get

1. 
$f(w)=\sqrt{w}$ 
$y=f(x)$ 

2. 
$g(r)=2r1$ 
$y=g(f(x))$ 

3. 
$q(s)=g(s)f(s)$ 
$y=q(x)$ 

4. 
$r(h)=0.1 \sin(100h) + e^{\cos h}$ 
$y=r(x)$ 

5. 
$s(p)=r(q(g(p)))$ 
$y=s(x)$ 

 Complete the following definition of the function L that uses
an xcoordinate, a ycoordinate, and a rate of change
as parameters and which defines a function passing through that point
with the given rate of change. Test your definition in GC. Type ctrlL
to get a subscript.
$L(x_0, y_0, m,x) = $
 Define a function D that turns a number of hours, minutes, and
seconds into a number of days. Your function should produce a result
similar to this in GC: D(32, 17, 22) = 1.345394
 Define these functions in GC: $$\begin{align}f(r)
&= r^22\\[1ex]g(s) &= \dfrac{s}{3} + 1\\[1ex]h(q)
&= f(2q)  g\left(f(q)\right)\end{align}$$
Explain how GC determines that h(2) = 12.33333
 Define a function h in GC any way you wish.
 enter the statements given below on separate lines.
 Click the n slider.
 How does the second statement create a line segment that connects
the point $(n, 0)$ with the corresponding point on h’s
graph? Hint: $x ‑ k$ gives the distance between the
values of x and k.
$y = h(x)$
$$x = n, \lefty\frac{h(n)}{2}\right <
\left\frac{h(n)}{2}\right$$
 Define the function q in GC as
$q(t) = t(t  1)(t  2)$. Explain how GC determines
that $q\left(q\left(q(3)\right)\right) = 1685040$.
More with Function Notation and GC
Here is an activity that will give you more practice with understanding
function notation to model actual situations and with using function
notation in GC.
CLICK HERE