1. P.3 Functions and Their Graphs

2. Objectives Use function notation to represent and evaluate a function.
Find the domain and range of a function.
Sketch the graph of a function.
Identify different types of transformations of functions.
Classify functions and recognize combinations of functions.

3. Function
A function must make an assignment to each number in the domain.
A function can assign only one number to any given number in the domain.
function
not a function

4. Domain and Range
Domain: the set of values that the independent variable (usually x) can take on
Range: the set of values for the dependent variable (usually y) if all numbers in the domain are plugged into the function

5. Implicit Form
Not explicitly solved for the variable
Explicit form

6. Function Notation

7. Example

8. Example

9. Domain and Range Domain is explicitly defined D: [4,5] R: [1/21, 1/12]
D: x≠±2 (implied)
R: f(x)≠0

10. Domain Restrictions Denominator can not be zero
Radicand of an even root has to be greater than or equal to zero.

11. Examples
Find the domain and range.

12. Graph of a Function How do you tell if a graph is a function?
Vertical line test No
Yes
Yes

13. Basic Functions Look at the graphs on page 22. y=x y=x2 y=x3 y=√x
y=sinx
y=cosx

14. Basic Transformations (p.23)
y=f(x)
y=f(x-c)
y=f(x+c)
y=f(x)-c
y=f(x)+c
y=-f(x)
y=f(-x)
y=-f(-x)
Shifts right
Shifts left
Shifts down
Shifts up
Reflects about x-axis
Reflects about y-axis
Reflects about origin

15. Elementary Functions Algebraic (polynomial, radical, rational)
Trigonometric
Exponential and logarithmic

16. Polynomial Functions
Although a graph of a polynomial function can have several turns, eventually the graph will rise or fall without bound as x moves to the right or left.

17. Polynomial Functions
an(leading coefficient): determines right or left behavior of the graph
an>0
an<0
EVEN degree
ODD degree

18. Combinations and Domains
sum: f(x)+g(x)
what's in both f and g
difference: f(x)-g(x)
product: f(x)g(x)
quotient: f(x)/g(x)
(as long as g(x)≠0)

19. Example f(x)=2x-3 and g(x)=√x+1 D: (-∞,∞) D: [-1,∞)
f(x)+g(x)=2x-3+√x+1
f(x)-g(x)=2x – 3 – √x+1
f(x)g(x)=(2x – 3)(√x+1)
f(x)/g(x)=(2x – 3)/(√x+1)
D:[-1,∞)
D:(-1,∞)

20. Composite Functions f◦g(x)=f(g(x)) g◦f(x)=g(f(x)) Domain:
look at the final answer AND any steps before simplifying

21. Examples
f(x)=√x and g(x)=x2+1 Find the domains of the composite functions.

22. Even and Odd Functions f(-x)=f(x) even (symmetric to y-axis)
f(-x)=-f(x) odd (symmetric to origin)
Symmetric to x-axis?
If it is a polynomial, you can just look at the powers of x.

23. Examples f(x)=x3-x f(-x)=(-x)3-(-x) =-x3+x =-f(x)
odd (symmetric to origin)
g(x)=1+cosx
g(-x)=1+cos(-x)
=1+cosx=g(x)
even (symmetric to y-axis)

24. Neither Even or Odd Of course, some functions are neither even or odd.
f(x)=x2+x+2
f(-x)=(-x)2+(-x)+2=x2 – x + 2

25. Homework P.3 (page 27) #3, 9, 27-31 odd 32, 33-37 odd 41-45 odd
61, 63, 69, 71