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.

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.

Example 1: -- Give the domain of each function. g(x) = 1 . b) h(x) = √(1 – r2) x – 7 so 1 – r2 > 0 x – 7 ≠ 0  x ≠ 7 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’

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’.

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 121. 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.

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 122-123 CE #1-13 all WE # 1-10 all