1. Properties of Functions
Pre-Calc Lesson 4.1
Properties of Functions
Function- A rule that assigns to every element in the
‘D’ – domain, one element from the ‘R’ – Range
Domain – the set of all ‘input’ values – (set of all x’s)
Range – the set of all outcomes -- (set of all y’s)
But to be classified as a function – every ‘x’ value can
only ever be paired with one and only one ‘y’ value.
Example – (2,3), (3,-4), (-5,2) - defines a function
(2,3), (3,-4), (2, - 5) - does not define a function
The ‘x’ value of ‘2’ has two different ‘y’ values attached to it!! Just can’t happen
and still be classified as a function.

2. Simply stated – the domain consists of all
possible ‘x’ values
When asked to list the domain of a function from its
‘rule’ -- our first thought should be – “ I can use all real
numbers.
Then work backwards – sometimes the rule will have
notation that can eliminate some possible real numbers
For instance – if a square root is involved in the rule, the
quantity under the radical must always be > 0 to be
classified as ‘real’.
Also if a fraction is involved with a variable in the
denominator, we know that the denominator can never = 0,
so we have to eliminate any such value from our possible
domain.

3. Example 1: -- Give the domain of each function.
g(x) = b) h(x) = √(1 – r2)
x – so 1 – r2 > 0
x – 7 ≠ 0  x ≠ remembering last chapter’s work  -1 < r < 1
When analyzing graphs of functions, the best way to
determine the domain and range of a function is to make
use of vertical and horizontal lines and ‘mentally’ slide
them back and forth through the graph.
Where a vertical line intersects a graph, all of the ‘x’ values
from your points of intersection, determine your domain. If
there is ever a gap or spot where the vertical line does not
intersect, then these are all ‘x’ values that are not in your
domain.
Same situation applies to a horizontal line with your ‘Range’

4. Use your book -- page 120 -- example #2.
In general classifications: The ‘domain’ value -- ‘x’– is
Called the ‘independent’ variable.
The range value – ‘y’ – is called the ‘dependent’ variable
Functions are a ‘subset’ of a more general class of
equations called Relations. --This describes any ‘rule’
that pairs the members of two sets (Domain and Range)
(or x and y)
A lot of relations are also functions.
But often relations are not functions, because you may
get a situation where more than one ‘y’ value attaches
itself to one unique ‘x’.

5. For instance  x = y2 -- could produce two points like this:
(9,3) and (9, -3) therefore we have two unique points
that share the same ‘x’- value. (just can’t happen)
When you are given a graph of a ‘relation’, we use
the vertical line test to determine whether that relation
is more specifically -- a function!
Look on page Investigate the two graphs located
2/3 of the way down the page. Which graph is not the
graph of a function?
Vertical line test: If ever a vertical line intersects a given
graph in more than one point at a time, then the graph
is not the graph of a function.

6. Examples:
Sketch the graph of each set. Tell if the graph is the
graph of a function. If it is, give the domain, range,
and zeros
{(x,y): y = 4 – x2 b) x = y2
Hw: pgs CE #1-13 all
WE # 1-10 all