1. 2.1 Functions and Their Graphs

2. In this lesson you will:
Represent relations and functions.
Graph and evaluate linear functions.

3. Warm Up

4. Recall the Cartesian Coordinate Plane
y-axis 10
Quadrant II 8
Quadrant I 6 4
Origin 2
-10 -8 -6 -4 -2 O 2 4 6 8 10
x-axis -2 -4
Quadrant III
Quadrant IV
y-coordinate -6 -8
(10,-8)
x-coordinate

5. A RELATION is a mapping or pairing of input values with output values
A RELATION is a mapping or pairing of input values with output values. Here are ways to represent this pairing.
Domain (x-input) 2 -1 5 7
Range (y-output) 7 8 10
MAPPING
(-1,8), (2,8), (5,7), (5,10), (7,10)
SET OF ORDERED PAIRS: 4 2 6 -2 -4 -6 8 10 -8
-10 x y
GRAPH
TABLE x y -1 8 2 5 7 10

6. FUNCTION
A function is a special type of relation in which each element of the domain is paired with exactly one element from the range.
Domain 3 2 5 8
Range 6 7 9
Range 8
Domain 3 1 4 8

7. What is the domain and range of the relation shown in the graph
What is the domain and range of the relation shown in the graph. Is a relation or a function? 4 2 6 -2 -4 -6 8 10 -8
-10 x y
Domain -4 -6 2
Range -4 2 6
Since one element of the domain is paired with two elements from the range. This is not a function.
Domain: {-6, -4, 2}
Range: {-4, 2, 4, 6}

8. Express the relation below as a mapping, a table and a graph, and state the domain and range. Is it a function?
TABLE
(-2,3), (4,1), (5,5), (7,-1), (7,8) x y -2 3 4 1 5 7 -1 8
MAPPING x 4 -2 5 7 y 1 -1 3 5 8
One element in the domain is paired with two of the range. This is not a function 4 2 6 -2 -4 -6 8 10 -8
-10 x y
GRAPH
A vertical line crossing two points also shows that this is not a function!
D={-2, 4, 5, 7}
R={-1, 1, 3, 5,8}

9. Determine if the graphs below are functions?4 2 6 -2 -4 -6 8 10 -8
-10 x y 4 2 6 -2 -4 -6 8 10 -8
-10 x y
NOTE: Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION!
NOTE: Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION

10. Determine if the graphs below are functions?4 2 6 -2 -4 -6 8 10 -8
-10 x y 4 2 6 -2 -4 -6 8 10 -8
-10 x y
Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION!
Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION

11. Determine if the graphs below are functions?4 2 6 -2 -4 -6 8 10 -8
-10 x y 4 2 6 -2 -4 -6 8 10 -8
-10 x y
Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION!
Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION

12. Determine if the graphs below are functions?4 2 6 -2 -4 -6 8 10 -8
-10 x y 4 2 6 -2 -4 -6 8 10 -8
-10 x y
Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION!
Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION
We will call this the VERTICAL LINE test. It will help us decide if a relation is a FUNCTION.

13. Function Notation
By naming a function “f”, you can write the function using function notation:
The symbol f(x) is read a “the value of f at x” or simply “f of x”.
Note that f(x) is another name for y.
The input values for x are the domain of f.
The output values for f(x) or y are the range of f.
Function names could be represented by other names than f, such as g(x), or even names like abs(x) or sin(x).

14. f(1)=2(1)+1=3 f(-3)=2(-3)+1=-5 f(k)=2(k)+1=2k+1
If f(x) = 2x +1
f(1)=2(1)+1=3
f(-3)=2(-3)+1=-5
f(k)=2(k)+1=2k+1
f(x+h)=2(x+h)+1=2x+2h+1
f( ) = 2( ) + 1

15. Evaluate the following functions with the given values:
g(x) = 2x + 1
and h(x)= 4x -1 2 2
4( ) -1
g(3) =
2( ) + 1 3
h(2) =
h(0) =
h(z) = 2
= 4(4)-1
= 6 + 1
= 7
= 16 -1
=15
g(-5) =
2( ) + 1 -5 = 2
4( ) -1
= -9
= 0 -1
=-1
g(k) =
2( ) + 1 k
= 2k + 1 2
4( ) -1 z
g(r+1) =
2( ) + 1
= 4z -1 2
r+1
= 2r
= 2r + 3

16. x y (x,y) -1 1 2 2( ) -2( )- 1 2( ) -2( )- 1 2( ) -2( )- 1
Evaluate the following equation for the given domain and then determine if it represents a function.
y = 2x - 2x - 1 2
D={-1, 0, 1, 2} 4 2 6 -2 -4 -6 8 10 -8
-10 x y x y
(x,y) -1 1 2
2x - 2x - 1 2 2
2( ) -2( )- 1 3 -1
(-1,3) 2
2( ) -2( )- 1
(0,-1) -1 2
2( ) -2( )- 1 1 -1
(1,-1) 2
2( ) -2( )- 1
(2,3) 2 3
We can guess that is a parabola and observe that all over the x values, it has only one corresponding y value. So IT IS A FUNCTION.
We can also perform the vertical line test and verify that it is a function because it crosses the curve a only one point.

17. y x (x-axis,y-axis) -1 1 2 2( ) -2( )- 1 2( ) -2( )- 1 2( ) -2( )- 1
Evaluate the following equation for the given domain and then determine if it represents a function.
x = 2y - 2y - 1 2
D={-1, 0, 1, 2} 4 2 6 -2 -4 -6 8 10 -8
-10 x y y x
(x-axis,y-axis) -1 1 2
2y - 2y - 1 2 2
2( ) -2( )- 1 3 -1
(3,-1) 2
2( ) -2( )- 1 -1
(-1,0) 2
2( ) -2( )- 1 -1
(-1,1) 1 2
2( ) -2( )- 1 3
(3,2) 2
We can guess that is a parabola that opens to the right but most values in the domain have two values in the range. So it is not a FUNCTION.
We can also perform the vertical line test and verify that it is NOT a function because it crosses the curve at MORE THAN ONE POINT.

18. DISCRETE VS CONTINUOUS
(-2,2), (2,5), (4,5), (8,-3), (10,8)
f(x) = x +3
This function works for any real values in the domain. It is a CONTINUOUS FUNCTION.
This is a function that only exist for the given values of its domain. IT IS A DISCRETE FUNCTION. 4 2 6 -2 -4 -6 8 10 -8
-10 x y 4 2 6 -2 -4 -6 8 10 -8
-10 x y

19. A function like f(x)=2x+1
Is a called a linear function.
Linear functions are of the form y=mx+b.
m and b are constants.
The graph of a linear function is a line.

20. LINEAR Functions (Equations)
Linear equations:
Not linear equation:
y = 2x + 5 1 1
4x – 3x + 2x – 3 = y 3 4 2
Exponents are different than 1.
5x- 2y = 6 1 1
5x = 2x + 5 1 1
2x - 4 = 4y + 6 1 1 1 1
4b + a = 10
y = 10 1
All are one or two variable equations with exponents always 1.

21. Is this relation a function?

22. Is this relation a function?

23. Is this relation a function?

24. Is this relation a function?

28. Homework odd, 49, 52