1. Functions and Their Graphs
Section P.3

2. After this lesson, you should be able to:
evaluate a function
recognize the graphs of linear, squaring, cubing, square root, absolute value, rational, sine and cosine functions and graph them on a calculator
find the domain and range of a function
use the graphing calculator to graph piecewise functions
describe transformations of the basic functions
find the zeroes of a function
determine if a function is even or odd

3. Definition of Function
Real-Valued Function of a Real Variable:
Let X and Y be sets of real numbers.
A real-valued function f of a real variable x
from X to Y is a correspondence that assigns to
each number x in X
exactly one number y in Y.
The domain of f is the set X.
is the image of x under f
and is denoted by f(x).
The number y
The range of f is a subset of Y
and consists of all images
of numbers in X.

4. Functions X Y f
domain
range x y

5. Common Forms for Describing Functions
MAT SPRING 2007
Common Forms for Describing Functions
Typically, x is the independent variable and y is the dependent variable.
Implicit form:
Explicit form (Solve for y in terms of x):
Function notation:

6. Example—Evaluating a Function
MAT SPRING 2007
Example—Evaluating a Function
For the function f defined by , evaluate each expression. a) b)

7. Example—Evaluating a Function (cont.)
MAT SPRING 2007
Example—Evaluating a Function (cont.)
For the function f defined by , evaluate each expression. c)

8. Domain and Range of a function
More simply stated, we can say
Domain
x-values (or input values)
Range
y-values (or output values)
Try thinking of a function as a machine with a “crank”. The domain values are those you put into the machine. Turn the crank and the range values are those that come out the machine.
So think about putting in domain values for x and getting out the corresponding range values for y (or f(x)). You’ve done this when you created tables of values.

9. Domain of a Function
The domain of a function may be described explicitly.
Example:
The domain may also be described implicitly. The implied domain is the set of all real numbers for which the function is defined.
Example: Define the domain of the function:

10. Interval Notation-Refresher
Open interval: (a, b)
Closed interval: [a, b]
(a, b]
[a, b)
Note: The interval is ALWAYS open where infinity is part of the interval notation.

11. Domain and Range of a Function
Example: Graph the function on your calculator, and then find the domain and range of the function.

12. Domain and Range--Piecewise Function
Example: Graph on your calculator and then determine the domain and range of the function.
You can find the inequality symbols using   to get the TEST menu

13. Evaluate a piecewise function
Example: For the piecewise function defined, evaluate each expression.

14. Vertical Line Test
To determine if a graph is a function, a vertical line test can be performed. In order for a graph to be a function, a vertical line in any spot on the graph can only intersect the graph at most once.

15. Basic Functions
The graphs of the eight basic functions from page 22 in your text should be recognizable to you from now on.
Absolute Value Function:
Identity Function:
Rational Function:
Squaring Function:
Cubing Function:
Sine Function:
Square Root Function:
Cosine Function:

16. Graphs of the Basic Functions (pt 1)
Identity Function
Squaring Function
Domain:
Range:

17. Graphs of the Basic Functions (pt 2)
Cubing Function
Square Root Function
Domain:
Range:

18. Graphs of the Basic Functions (pt 3)
Absolute Value Function
Rational Function
Domain:
Range:

19. Graphs of the Basic Functions (pt 4)
Sine Function
Cosine Function
Domain:
Range:

20. One-To-One Function
A function from X to Y is one-to-one if to each y-value in the range, there corresponds exactly one x-value in the domain.
Passes Horizontal Line Test

21. One-To-One Function The following functions are NOT one-to-one.
These graphs do NOT pass the Horizontal Line Test

22. Transformations of Functions
Let c > 0.
y = f(x)
Original graph
y = -f(x)
y = f(-x)
y = f(x – c)
y = f(x + c)
y = f(x) + c
y = f(x) - c

23. Transformations of Functions
Describe the transformations that the graph of f(x) must undergo to obtain the graph of g(x).
Orig:

24. Polynomial Functions
The most common algebraic function is the polynomial function.
n is a positive integer (the degree of the polynomial function)
ai are coefficients (an is the leading coefficient and a0 is the constant term)

25. Rational Functions The function f(x) is rational if
Polynomial, rational, and radical functions are all algebraic functions. The other types are trig and exponential functions. Non-algebraic functions are called transcendental functions.

26. Elementary Functions Elementary Functions Types Algebraic
Trigonometric
Exponential &
Logarithmic
Examples
Polynomial
Radical
Rational
Sine
Cosine
Tangent
Cotangent
Secant
Cosecant

27. Composite Functions Let f and g be functions. The function given by
(f ° g)(x) = f(g(x)) is called the composite of f with g.
The domain of f ° g is the set of all x in the domain of g such that g(x) is in the domain of f. x
f(g(x))
f ° g g f
g(x)

28. Example 1-Composite Functions
*Example: Given f(x) = x2 – 1 and g(x) = cos x, find f ° g.
(Pythagorean Identity cos2x+ sin2x = 1)

29. Example 2-Composite Functions
Example: Given and , find

30. Tests for Even and Odd Functions
The function y = f(x) is even if f(-x) = f(x)
(has y-axis symmetry)
The function y = f(x) is odd if f(-x) = -f(x)
(has origin symmetry)

31. Example of Odd and Even functions
Example Determine whether the function is even, odd, or neither.
a) b)

32. Homework
Section P.3: page 27 #1, 3, 7, 17, 19, odd, odd, 55, 59, 61, 93 (a and b only)