Algebra Unit
Functions Intro – part I
The Coordinate Grid
The x-axis runs horizontally
The y-axis runs vertically
The grid is divided into four quadrants
2 1
3 4
The Coordinate Grid
The points are always listed (x, y)
And called “an ordered pair”
(2, 3)
Functions can have positive function values or negative function values.
When the function is above the x -axis, it is positive.
When the function is below the x -axis, it is negative.
Relations
Every set of ordered pairs are called a “RELATION”
All relations are made up of a set of
x-values called the domain and a set of
y-values called the range.
One way to remember that is that x comes before y alphabetically
And d comes before r
Domain range
Some relations are also functions.
Functions are when you have a rule that describes the relationship between two variables
Each INPUT has only one OUTPUT
That’s important: If one input has two different outputs, then it’s NOT a function
More on that in a minute….
Functions are kind of like an equation
BUT we use f(x) INSTEAD of y (it’s pronounced “f of x”)
So where you might be used to seeing y = 2x + 7
When written as function, it would be f(x) = 2x + 7
“Do 2 times that x and add 5”
Say you have the function: f(x) = 2x + 5
It might help to think of functions a bit like a FACTORY or a MACHINE
that takes the input (x) and produces and output (y) or what we are now calling f(x)
How to tell if a set of ordered pairs is a function.
If a relation is a function, each unique x -value has a single unique y -value.
In other words every x can have ONLY one y value (1, 3)
This IS a function → (2, 5)
(3, 7)
(5, 11)
—-------------------------------------------------------------------------------------
This is NOT a function → (1, 3)
(2, 5)
(2, 7)
(5, 11)
Because the x -value of 2 has two different y -values: 5 and 8.
How to tell if a set of ordered pairs is a function.
If a relation is a function, each unique x -value has a single unique y -value.
(2, 3)
(4, 7)
(5, 9)
(7, 13)
Function Or Not?
(2, 3)
(4, 7)
(4, 9)
(7, 13)
Function Or Not?
(-2, 8)
(1, 4)
(2, 8)
(3, 13)
Function Or Not?
How to tell if a set of ordered pairs is a function.
If a relation is a function, each unique x -value has a single unique y -value.
(2, 3)
(4, 7)
(5, 9)
(7, 13)
Function Or Not?
Yes
f(x) = 2x-1
(2, 3)
(4, 7)
(4, 9)
(7, 13)
Function Or Not?
No! – the x value of 4 has two y values!
(-2, 8)
(1, 4)
(2, 8)
(3, 13)
Function Or Not?
Yes – it’s okay if there are y’s with two different x values
f(x) = x2 + 4
Solving simple functions:
If you are given f(x) = 4x-1 and are asked to find f(3)
Just substitute the 3 for the x in that equation
f(3) = 4(3) - 1 which equals 12-1 which = 11
We know that x is 3 because f(x) is f(3)
And we solved for f(x) to find 11 and since f(x) is y
We know the ordered pair is (3, 11)
Types of Functions
Linear functions are lines
f(x) = mx + b
Where m is the slope and b is the y-intercept
Types of Functions
Quadradic functions are curves
How can you tell?
The x is squared
You also have curves like this: