1. Chapter 2: Functions and Linear Functions

2. 2.1 Intro to Functions Relation: Set of ordered pairs
E.g., {(name, height)}
{(year, sales)}
{{year, Hawaii population)}
Domain: Set of all first components
E.g., {name}
{year}
Range: Set of all second components
E.g., {height}
{sales}

3. Functions
Function: set of ordered pairs such that each element of the domain corresponds to exactly one element in the range
Which is a function?
{(0,9.1), (10,6.7), (20,10.7), (30,13.2)}
{(1,5), (2,5), (3,7), (4,8)}
{{5,1), (5,2), (7,3,), (8,4)}

4. Function Notation y = 2x2 + 1 f(x) = 2x2 + 1 What is f(1)?
y is a function of x such that y = 2x2 + 1
f(x) = 2x2 + 1
“f of x equals 2x2 + 1”
What is f(1)?
f(1) = 2(1) = 3
What is f(3)?
f(3) = 2(3) = 19

5. Function Notation (cont)
Given:
f(x) = 2x2 + 1
What is f(-5)?
f(-5) = 2(-5) = 51
What is f(a + b)?
f(a + b) = 2(a + b) = 2(a2 + 2ab + b = 2a2 + 4ab + b2 + 1

6. Function Notation (cont)
Given:
g(x) = (3x – 1) / (x – 4)
What is g(2)?
g(2) = (3(2) – 1) / (2 – 4) = 5/(-2) = -5/2
What is g(-3)?
g(-3) = (3(-3) – 1) / ((-3) – 4) = (-9 – 1) / (-3 – 4) = -10 / = 10 / 7

7. 2.2 Graphs of Functions Given: Graph of f(x): f(x) = 2x + 4
Graph of {(x, y) | y = 2x + 4} x y

8. Vertical Line Test
Is this a function?
f(x)
g(x)

9. Find f(x) from Graph f(3) = ? Domain of f(x) ? Range of f(x) ? 20 10-2 -1 1 2 3 4
-10
Domain of f(x) ?
Range of f(x) ?
-20

10. 2.3 Algebra of Functions Given: f(x) = 2x + 1 g(x) = x - 1
What is: (f + g)(x)? (f + g)(x) = f(x) + g(x) = (2x + 1) + (x – 1) = 3x
(f + g)(x)
g(x)
f(x)

11. Algebra of Functions (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x)

12. Your Turn Given: f(x) = x2 – 3 g(x) = 4x + 5 Find: (f – g)(x)

13. 2.4 Linear Function & Slope
Given:
y = 2x + 1
When x = 0, y = 1
For every increase in x, y increases by 2x.
Δy = 2
(0, 1)
Δx = 1

14. Linear Function & Slope
Given:
y = mx + b
Y-intercept y
Slope: m
Δy = m
(0, b)
Δx = 1

15. Your Turn Sketch the graph of the following equations.
4x – 3y = 6
3x = 5y – 15
What is the slope of the line determined by the following points
(3, 1) & (5, 4)
(-6, -3) & (4, - 3)

16. 2.5 Slope-Intercept Form of Line
Given a line:
y-intercept = 5
Slope = 3/4
Find the equation of the line
(y – 5)/(x – 0) = 3 / 4 4(y – 5) = 3(x) 4y – 20 = 3x 4y = 3x + 20 y = (3/4)x + 5
(x, y) Δy
(0, 5) Δx
(Δy / Δx) = 3 / 4

17. 2.5 Point Slope Form for Line
Given:
Line passes through (1, 2) & (3, 5)
Find the equation of the line
m = (5 – 2) / (3 – 1) = 3 / 2 (y - 2) / (x – 1) = 3 / 2 2(y – 2) = (3)(x – 1) 2y – 4 = 3x – 3 2y = 3x + 1 y = (3/2)x + 1/2
(x, y)
(3, 5) Δy
(1, 2) Δx