1.2
Function Notation
Given an
equation for a function, e.g. 
, or 
we can let 
replace the
expression in x. In this way, we
can speak about functions in general as being of the form 
, without having to specify the particular expression in x. Furthermore, this allows us to name a
function using a letter. Thus, if we
speak of the function ‘g’, we mean that there is some expression 
and an equation 
where 
gives the
‘instructions’ on how to compute the output number y from the input
number x.
For example, if
we speak of the function 
, then we know that the equation of the function is 
, and we know that we find the output number corresponding to
a given input number by adding one to the square of the input number, then
dividing by two.
Now, one might
ask, if we already have 
, why do we need to write 
? The answer is: we don’t have to. We can think of 
itself as
representing the output number corresponding to the input number when x
is put into the function h. Now keep
in mind that we are now thinking of 
in two different
ways. We are thinking of it both
as representing the expression 
, and as representing the output of the function h
when x is the input.
Exercise
1.2.1
Make a table
containing two columns. Label the first
column x and the second column
. Pick five values
for x and place them in the first column. Pick some decimal values as well as whole number values, and
negative as well as positive values.
Pick all input values of x such that their absolute values are
not larger than 2. Now, compute the
corresponding output values of 
using 
and place the values
in column 2. Plot each resulting pair of
numbers (x,y) in the Cartesian plane, where 
.
Notice that 
, 
, etc. So whatever
number is place between the parentheses in the expression 
must replace each occurrence
of x in the expression 
. To make this more
explicit, we could write the function h in blank parenthesis form
as follows 
. Then any number
placed into the blank parenthesis following the h must also be placed
into the blank parenthesis on the right side of the equation. The next example will show why this is an
important idea.
Let us suppose
that the input number x is itself computed from a number t
according to the formula 
. Then, since 
, it follows that 
. Thus, not only can
we place numbers into the blank parentheses in 
, but we can place algebraic expressions as well.
Exercise
1.2.2
Find 
, 
, and 
. Simplify each
expression.
Exercise
1.2.3
Find the value
of 
and simplify, given
that 
.