1. 1.3 Graphs of Functions
Pre-Calculus

2. Home on the Range What kind of "range" are we talking about?
What does it have to do with "domain?"
Are domain and range really "good fun for the whole family?"

3. Definitions
Domain: Is the set of all first coordinates (x-coordinates) from the ordered pairs.
Range: Is the set of all second coordinates (y-coordinates) from the ordered pairs.

4. Domain
The domain is the set of all possible inputs into the function { 1, 2, 3, … }
The nature of some functions may mean restricting certain values as inputs

5. Range
{ 9, 14, -4, 6, … }
The range would be all the possible resulting outputs
The nature of a function may restrict the possible output values

6. Find the Domain and Range
Given the set of ordered pairs,
{(2,3),(-1,0),(2,-5),(0,-3)}
Domain
Range

7. Choosing Realistic Domains and Ranges
Consider a function used to model a real life situation
Let h(t) model the height of a ball as a function of time
What are realistic values for t and for height?

8. Choosing Realistic Domains and Ranges
By itself, out of context, it is just a parabola that has the real numbers as domain and a limited range

9. Find the Domain and Range of a Function
Find the domain of f(x)
Find f(-1)
f(2)
Find the range of f(x)
*When viewing a graph of a function, realize that solid or open dots on the end of a graph mean that the graph doesn’t extend beyond those points. However, if the circles aren’t shown on the graph it may be assumed to extend to infinity.

10. Domain and Range
Find the domain and range of

11. Vertical Line Test
A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

12. If a vertical line passes through a graph more than once, the graph is not the graph of a function.
Hint:
Pass a pencil across the graph held vertically to represent a vertical line.
The pencil crosses the graph more than once. This is not a function because there are two y-values for the same x-value.

19. The Ups and Downs
Think of a function as a roller coaster going from left to right
Uphill
Slope > 0
Increasing function
Downhill
Slope < 0
Decreasing function 19

20. Increasing/Decreasing Functions
A function f is increasing on (a, b) if f (x1) < f (x2) whenever x1 < x2.
A function f is decreasing on (a, b) if f (x1) > f (x2) whenever x1 < x2.
Increasing
Decreasing
Increasing

21. Example
In the given graph of the function f(x), determine the interval(s) where the function is increasing, decreasing, or constant.

22. Maximum and Minimum Values
Absolute Maximum
( f (c1)  f (x) for all x)
Local Maximum
( f (c2)  f (x) for all x in I • •
| c1
| c2 I

23. Maximum and Minimum Values
Absolute Minimum
( f (c1)  f(x) for all x)
c2 |
| c1 • I I •
Collectively, maximums and minimums are called extreme values.
Local Minimum
( f (c2)  f(x) for all x in I )

24. Approximating a Relative Minimum
Use a calculator to approximate the relative minimum of the function given by

25. Approximating Relative Minima and Maxima
Use a calculator to approximate the relative minimum and relative maximum of the function given by

26. Temperature
During a 24-hour period, the temperature y (in degrees Fahrenheit) of a certain city can be approximated by the model where x represents the time of day, with x=0 corresponding to 6 am. Approximate the max and min temperatures during this 24-hour period.

27. Piecewise Defined Functions
Sketch the graph of
by hand.

28. Even functions A function f is an even function if
for all values of x in the domain of f.
Example: is even because

29. Odd functions A function f is an odd function if
for all values of x in the domain of f.
Example: is odd because

30. Determine if the given functions are even or odd

31. Graphs of Even and Odd functions
The graph of an even function is symmetric with respect to the x-axis.
The graph of an odd function is symmetric with respect to the origin.

32. Determine if the function is even or odd?

33. Determine if the function is even or odd?

34. Determine if the function is even or odd?

35. Homework
Page 38-41
2-8 even (graphical), all, odd, even, 44, 48, odd, odd, 91