1. 2.3 – Functions and Relations
Relations, Domain and Range
Defn: A relation is a set of ordered pairs.
Domain: The values of the 1st component of the ordered pair.
Range: The values of the 2nd component of the ordered pair.

2. 2.3 – Functions and Relations
Relations, Domain and Range
State the domain and range of each relation. x y 1 3 2 5 -4 6 4 x y 4 2 -3 8 6 1 -1 9 5

3. 2.3 – Functions and Relations
Defn: A function is a relation where every x value has one and only one value of y assigned to it.
State whether or not the following relations could be a function or not. x y 4 2 -3 8 6 1 -1 9 5 x y 1 3 2 5 -4 6 4 x y 2 3 5 7 8 -2 -5
function
not a function
function

4. 2.3 – Functions and Relations
State whether or not the following mappings of two relations could be a function or not. x y x y
Relation A is not a function.
Relation B is a function.
For x = 1, there are two y values (3 and 4).
For each value of x, there is one value of y.

5. 2.3 – Functions and Relations
Graphs can be used to determine if a relation is a function.
Vertical Line Test
If a vertical line can be drawn so that it intersects a graph of an equation more than once, then the equation is not a function.

6. 2.3 – Functions and Relations
The Vertical Line Test y
function x y -3 5 7 -2 -7 4 3 x

7. 2.3 – Functions and Relations
The Vertical Line Test y
not a function x y 1 -1 4 2 -2 x

8. Domain and Range from Graphs
2.3 – Functions and Relations
Functions
Domain and Range from Graphs x y
Find the domain and range of the function graphed to the right. Use interval notation.
Domain
Range
Domain:
[–3, 4]
Range:
[–4, 2]

9. Domain and Range from Graphs
2.3 – Functions and Relations
Functions
Domain and Range from Graphs x y
Find the domain and range of the function graphed to the right. Use interval notation.
Range
Domain:
(– , )
Range:
[– 2, )
Domain

10. 2.3 – Functions and Relations
Function Notation
Shorthand for stating that an equation is a function.
Defines the independent variable (usually x) and the dependent variable (usually y).

11. 2.3 – Functions and Relations
Function Notation
Function notation also defines the value of x that is to be use to calculate the corresponding value of y.
f(x) = 4x – 1
find f(2).
g(x) = x2 – 2x
find g(–3).
find f(3).
f(2) = 4(2) – 1
g(–3) = (-3)2 – 2(-3)
f(2) = 8 – 1
g(–3) = 9 + 6
f(2) = 7
g(–3) = 15
(2, 7)
(–3, 15)

12. 2.3 – Functions and Relations
Function Notation
Given the graph of the following function, find each function value by inspecting the graph. x y ●
f(x) ●
f(5) = 7
f(4) = 3 ●
f(5) = 1
f(6) = 6 ●

13. 2.4 – linear Equations in Two Variables and Linear Functions
Three Forms of an Equation of a Line
Slope-Intercept Form:
This form is useful for graphing, as the slope and the y-intercept are readily visible.
Point-Slope Form:
The point-slope form allows you to use ANY point, together with the slope, to form the equation of the line.
Standard Form:
Note: A, B, and C cannot be fractions or decimals.

14. 2.4 – linear Equations in Two Variables and Linear Functions
Slope of a Line
Slope formula =
Green slant line: positive slope
Vertical line: undefined or no slope
Horizontal line: slope = 0
Δ𝑦=0
Red slant line: positive slope
Δ𝑥=0
Constant Functions
Linear Functions

15. 2.4 – linear Equations in Two Variables and Linear Functions
Slope of a line
Slope formula =
Find the slope of the line containing the following points:
−3, 7 𝑎𝑛𝑑 1, 4
𝑚= 7−4 −3−1
𝑚= 3 −4
=− 3 4

16. 2.4 – linear Equations in Two Variables and Linear Functions
Slope-Intercept Form: 𝑦=𝑚𝑥+𝑏
𝑚:𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑏:𝑡ℎ𝑒 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 0, 𝑏
𝑦= 8 11 𝑥−6
5𝑥+2𝑦=7
2𝑦=−5𝑥+7
𝑚= 8 11
𝑦=− 5 2 𝑥+ 7 2
𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0, −6
𝑚=− 5 2
𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0, 7 2

17. 2.4 – linear Equations in Two Variables and Linear Functions
The Average Rate of Change
If f is defined on the interval [x1, x2], then the average rate of change
of f on the interval [x1, x2] is the slope of the secant line containing
(x1, f(x1)) and (x2, f(x2)).
Average rate of change:

18. 2.4 – linear Equations in Two Variables and Linear Functions
Determine the average rate of change
From (–1.5, 0) to (0, 2)
𝑚= 𝑓 0 −𝑓 −1.5 0− −1.5
𝑚= 2−0 0− −1.5
𝑚= 2 1.5
𝑚= 4 3
From (0, 2) to ( 2.5, 1)
𝑚= 𝑓 2.5 −𝑓 −0
𝑚= 1−2 2.5−0
𝑚= −1 2.5
𝑚=− 2 5

19. Solving Equations and Inequalities Graphically
2.4 – linear Equations in Two Variables and Linear Functions
Solving Equations and Inequalities Graphically
𝑦=𝑥+5 𝑎𝑛𝑑 𝑦=−3𝑥−3
𝑥+5=−3𝑥−3 •
4𝑥+5=−3
4𝑥=−8
𝑥=−2
𝑥+5>−3𝑥−3
−2, ∞
𝑥+5≤−3𝑥−3
−∞, −2

20. Solving Equations and Inequalities Graphically
2.4 – linear Equations in Two Variables and Linear Functions
Solving Equations and Inequalities Graphically
𝑦=−4𝑥+4 𝑎𝑛𝑑 𝑦=2𝑥−2
−4𝑥+4=2𝑥−2
−6𝑥+4=−2
−6𝑥=−6 •
𝑥=1
−4𝑥+4≥2𝑥−2
−∞, 1
−4𝑥+4≤2𝑥−2
1, ∞