Function 1-7
Function: a
relation (set of ordered pairs) where there is EXACTLY ONE output for each
input. Each element of the domain has EXACTLY ONE element in the range. THE X
VALUES NEVER REPEAT!
Vertical Line
Test: If the relation is represented with a GRAPH, this test is the
easiest way to see if an x value repeats. Draw vertical lines up and down
continuously on the graph and see if a line intersects with (hits) more than
one point. If it does, it’s a relation, but not a function. If it doesn’t, it’s
a function.
***Discrete function: a function where the ordered pairs are not connected (For example, you can’t purchase a part of a candy bar at 7-11)
***Continuous function: a function where the ordered pairs are connected in a smooth curve (For example, if you’re driving in a car, the distance ever increases continuously)
Function notation:
f(x): If a relation is a function, you can write the equation using y as a
variable as the function value OR you can use f(x) as the function value. You
read this as “the function of x” or the function value for the given x
value. Note that “f” is NOT A VARIABLE…it’s an abbreviation for the
word FUNCTION…so don’t ever divide by f
Example: y = 2x + 3 OR f(x) = 2x +
3 represent the same function.
To find f(2) in
the above function, simply plug in 2 for x and evaluate: f(2) = 2(2) + 3 = 7 so
f(2) = 7
WHAT’S
BETTER ABOUT f(2)=7 vs y=7 although they mean the same thing? In function notation, you know both the
domain value and the range value! Using
other letters with function notation:
Another good thing
about function notation is that you can use specific letters that show the
relationship between two variables. For example, the cost of what you spend
depends on how much you buy. Say you’re only buying pizzas for a big party. Let
c represent the cost of the pizza and p represent the number of pizzas you
purchase. The function notation of c(p) would be expressed in words as “the
cost of the pizza”. Notice: the variable inside the ( ) is the
input/independent variable/domain and the outside variable is the
output/dependent variable/range. … What you spend depends on the number of
pizzas you order!
You can
multiply or divide functions. The way you express this is to simple show the
operations on the f(x): 2[f(x)] means
to double the function Example: If f(x)
= 2x + 3 then 2[f(x)] = 2(2x + 3) = 4x
+ 6
LINEar
function: A set of ordered pairs that draws a straight line (that’s not
vertical)
NonLINEar function: A set of ordered pairs that does NOT draw a straight line
