MORE PRACTICE WITH FUNCTION NOTATION

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   $\,x\,$ ’.

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:
$f(-3) = (-3)^2 + 2(-3) = 9 - 6 = 3$
Question:
Let $\,f(x) = x^2 + 2x\,$.
Find and simplify:   $f(x+1)$
Solution:
$f(x+1) = (x+1)^2 + 2(x+1) = x^2 + 2x + 1 + 2x + 2 = x^2 + 4x + 3$

## 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 $$f(\text{blah}) = (\text{blah})^2 + 2(\text{blah})$$ or, even more simply, just leave a blank space for the input—a pair of empty parentheses where the input should be: $$f(\ \ \ \ ) = (\ \ \ \ )^2 + 2(\ \ \ \ )$$ Then, when you want to find (say) $\,f(x+1)\,$, just put the input, $\,x+1\,$, inside every set of empty parentheses: $$f(x+1) = (x+1)^2 + (x+1)$$ 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:
$f(1) + f(3)$
$\ \ = \overset{f(1)}{\overbrace{(1^2 - 2\cdot 1)}} + \overset{f(3)}{\overbrace{(3^2 - 2\cdot 3)}}$
$\ \ = (1 - 2) + (9- 6)$
$\ \ = 2$
Master the ideas from this section