﻿ More Practice with Function Notation
MORE PRACTICE WITH FUNCTION NOTATION
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

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

When you're done practicing, move on to:
Domain and Range of a Function

CONCEPT QUESTIONS EXERCISE: