1. 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 output values for the dependent variable (usually y), resulting from members of the domain being plugged into the function.

5. Explicit and Implicit Form
Given that y is represents f(x) for a function.
Explicit form is solved for y.
example:
Implicit form shows the relationship between y and x but is not solved explicitly for y.

6. Function Notation

7. Example:
Lets take a closer look at this!

8. Example:
An example of this type is referred to as a difference quotient.
The difference quotient will be used extensively in this class!

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

10. Domain Restrictions: Denominator can not be zero
Radicand of an even root has to be greater than or equal to zero.
Argument of a logarithm must be positive.
It is extremely important to consider domain restrictions throughout Calculus!

11. Examples
Find the domain and range of each function:

12. Graph of a Function How do you tell if a graph is a function?
One input value causes one output value.
Vertical line test No
Yes
Yes

13. Basic Functions: Square root function: Identity function:
Squaring function:
Cubing function:
Absolute value function:
Rational function:
Sine function:
Cosine function:

14. Basic Transformations of y=f(x) c>0
y=f(x-c) f(x) shifted right c units.
y=f(x+c) f(x) shifted left c units.
y=f(x)-c f(x) shifted down c units.
y=f(x)+c f(x) shifted up c units.
y=-f(x) f(x) reflected about x.
y=f(-x) f(x) reflected about y.
y=-f(-x) f(x) reflected about the origin.

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

16. Polynomial Functions
Degree: the highest power on a polynomial, n, right and left behavior depends on whether degree is odd or even.
Lead coefficient: Coefficient of the highest degreed polynomial term; denoted an.
Constant: Coefficient of the zero degreed polynomial term.
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 given an odd or even degree.
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
Evaluate each expression below stating the domain and range of each:

20. Composite Functions f◦g(x)=f(g(x))
The domain of f(g(x)) is any x value such that both of the following statements are true:
- x is in the domain of g.
and
- g(x) is in the domain of f.
In short: look at the domain of the simplified composite function AND of the function used as input into the composite function.

21. Examples Evaluate and state the domain of the composite function .

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. Example: f(x)=x2+x+2
f(-x)=(-x)2+(-x)+2=
x2 – x NOT EVEN
-f(-x)=-((-x)2+(-x)+2)=
- x2 + x – 2 NOT ODD
The function is neither even or odd.

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