Function Notes
1.6 Functions
To show a relation (a set of ordered pairs), you can use:
ordered pairs Table Graph Mapping
· A function is:
· a set of ordered pairs in which each x value is paired with only one y.
OR
· no two ordered pairs have the same x value.
Independent & Dependent variables –
Relationship with a dependency.
Ex. Calculators depend on Batteries.
Independent – batteries
Dependent – calculator
Ex. Flowers depend on water to grow.
Independent – water
Dependent – flowers
Domain & Range -
Terms used for x & y variables
Domain is independent (x)
Range is dependent (y)
Input & Output –
Input is independent – it doesn’t matter what you put in the mix.
Output is dependent – the output depends on what you put in the mix.
Is the relation a function?
From Tables:
*Check the x values (domain/input). If they don’t repeat, it IS a function.
1.
Input (x) / 1 / 2 / 3 / 4 / 5Output(y) / 5 / 4 / 3 / 2 / 1
Domain {1,2,3,4,5}
Range {5,4,3,2,1}
2.
Input (x) / 1 / 2 / 3 / 4 / 5Output(y) / 4 / 4 / 4 / 4 / 4
Domain
Range
3.
Input (x) / 1 / 1 / 3 / 4 / 5Output(y) / 5 / 4 / 3 / 2 / 1
Domain
Range
From Ordered Pairs:
*Check the x values (domain/input). If they don’t repeat, it IS a function.
4. (3,2) (4,5) (5,4) (6,1) (2,5)
5. (1,2) (1,3) (1,4) (1,5) (1,6)
From Graphs
Vertical Line test – Draw a vertical line on graph.
· If it crosses the relation ONLY once it is a function.
· If it crosses the relation MORE than once it is NOT a function.
YES NO YES
Is the graph a function?
*use the line test.
YES YES NO
From Maps
The input (x) can only have one arrow leaving each number. The output (y) can have more than one arrow going to it.
Input Output Input Output Input Output
Function Function NOT a Function
Given the Domain, What is the Range of the function?
Make a table then plug in the domain values for the x and find the range (y) values.
Ex: Y = 3x + 4
Domain: {0, 5, 7, 10}
X / Y0 / 4
5 / 19
7 / 25
10 / 34
X / Y
0
5
7
10
Process
Range: {4, 19, 25, 34}
To graph a function from a table
· make ordered pairs to plot on the graph.
x / y0 / 0
1 / 2
2 / 4
3 / 6
4 / 8
Ordered Pairs
(0,0)
(1,2)
(2,4)
(3,6)
(4,8)
To graph a function from an equation
· make a table, fill in the domain (x) (make up numbers for x if a domain is not given) and find (do the process) the range (y) –. Make ordered pairs to plot on the graph.
EX: Y = 2x + 1
With a domain of {-2, -1, 0, 1, 2}
x / Process / y-2 / 2(-2)+1 / -3
-1 / 2(-1)+1 / -1
0 / 2(0)+1 / 1
1 / 2(1)+1 / 3
2 / 2(2)+1 / 5
Ordered Pairs
(-2,-3)
(-1,-1)
(0,1)
(1,3)
(2,5)
5.6 Scatter Plot Correlations
Positive correlation – Positive movement (to the right & up) positive x & positive y.
Negative correlation – negative movement (to the right & down) Positive x & NEGATIVE y.
No correlation – no trend. Points randomly placed.
Make a prediction from the scatter plot.
Making Predictions
4.2 (Extension) Discrete & Continuous Functions
Discrete Function has a graph with isolated points.
Domain: x = 1, 2, 3, 4, 5
Continuous Function has a graph that is unbroken.
Domain: x 0
4.7 Function Notation
v If something is a function, its equation can be written in a form called function notation.
v Ex: The equation y = 3x + 4 is a function.
§ It can be rewritten as f(x) = 3x + 4.
§ All you do is substitute the y for the f(x).
§ The equation f(x) = 3x + 4 is read “f of x equals 3x + 4.”
§ ‘F of x’ stands for ‘a function of x’.
§ You do not have to use the letter “f” to write a function. You may use any variable.
Examples: g(x) = x – 1, t(x) = 15 – 3x.
v You can evaluate equations in function notation.
§ Ex 1: Given f(x) = 3x + 4. Find f(2).
§ You replace x with 2 in the right side of the equation and evaluate (find an answer.)
f(2) = 3(2) + 4
f(2) = 6 + 4
f(2) = 10
§ Ex 2: Given f(x) = 3x + 4 find f(-5):
f(-5) = 3(-5) + 4
f(-5) = -15 + 4
f(-5) = -11
(The number inside the parenthesis is the value you substitute for the x to find your answer.)
Ex 3: Given h(x) = -2x – 3 Find h(4).
h(4) = -2(4) – 3
h(4) = -8 – 3
h(4) = -11