This document discusses piecewise functions and even and odd functions. It defines a piecewise function as a function with multiple sub-functions where each applies to a certain interval of the domain. It provides an example of finding values of a piecewise function at specific x values. It also outlines the steps to graph a piecewise function. The document then defines even functions as those where f(x) = f(-x) and odd functions as those where -f(x) = f(-x). It provides examples of even and odd functions and notes that most functions are neither. It concludes with some properties of even and odd functions.

1.  PIECE-WISE FUNCTION
Cislyn S. Semillano
MAED - MATHEMATICS
 ODD AND EVEN FUNCTION

2. TOPIC :
A piecewise function or a compound function is a function f(x) defined by
multiple sub-functions, where each sub-function applies to a certain
interval of the main function’s domain. It has different definitions in
different intervals of x. The graph of a piecewise function has different
pieces corresponding to each of its definitions.
Example #1: 𝐼𝑓 𝑓(𝑥)
Find: a. f(-4)
b. f(3)
5𝑥 + 2 , 𝑖𝑓 𝑥 ≥ 0

3. 𝐼𝑓 𝑓(𝑥)
5𝑥 + 2 , 𝑖𝑓 𝑥 ≥ 0

4. How to Graph Piece-wise Function:
1. Understand what each definition of the function represents. Write the intervals that are
shown in the definition of the function along with their definitions.
2. Make a table with two columns labeled x and y corresponding to each interval. Include the
endpoints of the interval without fail. If the endpoint is excluded from the interval then note
that we get an open dot corresponding to that point in the graph.
3. In each table, take more numbers (random numbers) in the column of x that lie in the
corresponding interval to get the perfect shape of the graph. If the piece is a straight line, then
2 values for x are sufficient. Take 3 or more numbers for x if the piece is NOT a straight line.
4. Substitute each x value from every table in the corresponding definition of the function to get
the respective y values.
5. Plot all the points from the table (taking care of the open dots) in a graph sheet and join them
by curves.

5. Graph: 𝑓(𝑥)
5𝑥 + 2 , 𝑖𝑓 𝑥 ≥ 0

6. EXAMPLE #2:
f(𝑥)
3𝑥+1 , 𝑖𝑓 𝑥 > 0
x y
-1 0
-2 -3
-3 -8
x y
1 4
2 7
3 10
3𝑥+1 , 𝑖𝑓 𝑥 > 0

7. TOPIC :Even and Odd Functions
Even Functions
A function is "even" when:
f(x) = f(−x) for all x
In other words there is symmetry about the y-
axis (like a reflection)
Example: f(x) = x2+1
Special types of functions
Odd Functions
A function is "odd" when:
−f(x) = f(−x) for all x
Note the minus in front of f(x): −f(x).
And we get origin symmetry.
Example: f(x) = x3−x

8. Neither Odd nor Even
Don't be misled by the names "odd" and "even" ... they are just names ... and a
function does not have to be even or odd.
In fact most functions are neither odd nor even. For example, just adding 1 to the
curve gets this:
Example: f(x) = x3−x+1
EVEN ODD

9. EXAMPLES:

10. EXAMPLES:

11. EXAMPLES:
By a quick comparison, it this doesn’t match, so this function is not even. To check, write down the exact opposite
of the original function, but with all of the signs changed: −f (x) = −2x3 + 3x2 + 4x − 4
This doesn't match what I came up with, either. So the original function isn't odd, either.
f (x) is neither even nor odd.

12. Do you know?
The only function that is even and odd is f(x) = 0
The sum of two even functions is even
The sum of two odd functions is odd
The sum of an even and odd function is neither even nor odd (unless one
function is zero).
The product of two even functions is an even function.
The product of two odd functions is an even function.
The product of an even function and an odd function is an odd function

13. THANK YOU
for listening 
“The proper function of a man is to
live, not to exist. I shall not waste
my days in trying to prolong them.
I shall use my time – John London”.