Recall from Introduction to Function Notation
that 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 $\,f\,$,
and the input name is $\,x\,$,
then the unique corresponding output is called $\,f(x)\,$.
The notation ‘
$f(x)\,$’ is read aloud as:
‘ $\,f\,$ of $\,x\,$ ’.
So, what exactly is $\,f(x)\,$?
Answer:
It is the output from the function $\,f\,$ when the input is $\,x\,\,$.
This exercise gives more advanced practice with function notation.
EXAMPLES:
Question:
Let $\,f(x) = x^2 + 2x\,$.
Find and simplify: $f(-3)$
Solution:
$\cssId{s20}{f(-3)}
\cssId{s21}{= (-3)^2 + 2(-3)}
\cssId{s22}{= 9 - 6}
\cssId{s23}{= 3}$
Question:
Let $\,f(x) = x^2 + 2x\,$.
Find and simplify: $f(x+1)$
Solution:
$\cssId{s28}{f(x+1)}
\cssId{s29}{= (x+1)^2 + 2(x+1)}
\cssId{s30}{= x^2 + 2x + 1 + 2x + 2}
\cssId{s31}{= x^2 + 4x + 3}$
the ‘Empty Parentheses Method’
Some people find it helpful to use the so-called ‘empty parentheses method’ to
help with function evaluation.
For example, take the function rule $\,f(x) = x^2 + 2x\,$ and rewrite it as
$$\cssId{s35}{f(\text{blah})}
\cssId{s36}{= (\text{blah})^2 + 2(\text{blah})}$$
or, even more simply,
just leave a blank space for the input—a pair of empty parentheses
where the input should be:
$$\cssId{s39}{f(\ \ \ \ )}
\cssId{s40}{= (\ \ \ \ )^2 + 2(\ \ \ \ )}$$
Then, when you want to find (say) $\,f(x+1)\,$,
just put the input, $\,x+1\,$, inside every
set of empty parentheses:
$$\cssId{s43}{f(x+1)}
\cssId{s44}{= (x+1)^2 + 2(x+1)}$$
Voila!
Question:
Let $\,f(x) = 5\,$.
Find and simplify: $f(x+1)$
Solution:
The function $\,f\,$ is a constant function:
no matter what the input is, the output is $\,5\,$.
That is, $\,f(\text{anything}) = 5\,$.
So, $\,f(x+1) = 5\,$.
Question:
Let $\,f(x) = x^2 - 2x\,$.
Find and simplify: $f(1) + f(3)$
Solution:
$\cssId{s58}{f(1) + f(3)}$
$\ \ \cssId{s59}{= \overset{f(1)}{\overbrace{(1^2 - 2\cdot 1)}}}
\cssId{s60}{+ \overset{f(3)}{\overbrace{(3^2 - 2\cdot 3)}}}$
$\ \ \cssId{s61}{= (1 - 2) + (9- 6)}$
$\ \ \cssId{s62}{= 2}$
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Domain and Range of a Function