8/24/2021
Even-Odd Function
Dolfus G. Miciano
Even Function
• Given f(x), if changing x into –x so that f(-x) = f(x) then f(x) is even.
• Example 1: Determine if f(x) = 4x4 - 3x2 + 3 is even or odd.
• Solution: Replacing x with – x
 f(-x) = 4(-x)4 - 3(-x)2 + 3
; simplify each term
4
2
= 4x - 3x + 3
; compare to the given
f(-x) = f(x)
Therefore, f(x) is an even function.
1
8/24/2021
Odd Function
• Given f(x), if changing x into –x so that f(-x) = - f(x) then f(x) is an odd
function.
• Example 2: Show f(x) = x5 + 3x3 - 3x is an odd function.
• Solution: Replacing x with – x
 f(-x) = (-x)5 + 3(-x)3 – 3(-x)
; simplify the terms
5
3
= - x - 3x + 3x
= - (x5 + 3x3 – 3x)
; factor out the negative
 f(-x) = - f(x)
Therefore, f(x) is an odd function.
Example 3: Determine if f(x) = sin (4x) is even or odd function
• Solution: Replacing x with – x
 f(-x) = sin(4(-x))
= sin(-4x)
= - sin(4x)
 f(-x) = - f(x)
; simplify the argument
; recall sin (-x) = - sin x
Therefore, f(x) is an odd function.
2
8/24/2021
Example 4: Is f(x) = -x3 + 5x – 4 an odd function?
• Solution: Replacing x with – x
f(-x) = -(-x)3 + 5(-x) – 4
= x3 - 5x – 4
= - (-x3 + 5x + 4)
f(-x) ≠ - f(x)
Also, f(-x) ≠ f(x)
; simplify the terms
; factor out the negative
Therefore, f(x) is neither odd nor even function.
Piecewise Function
Dolfus G. Miciano
3
8/24/2021
Piecewise Function
y
• A function defined by many conditions such as
f(x) =
2𝑥 − 1 𝑖𝑓 𝑥 < 1
2
𝑖𝑓 𝑥 = 1
𝑥 + 2 𝑖𝑓 𝑥 > 1
f(x)=2
f(x)=x+2
8
Tips: trace the graph by a
vertical line from left to right
and if there is no graph (gap),
then that set of numbers (gap)
is not included in the domain.
For the range use the horizontal
line by tracing from bottom
going up.
-9
-8
-7
-6
-5
f(x)=2x-1
-4
-3
-2
6
Y=x+2
4
(1, 2)
2
x
-1
1
-4
-6
 D= all real nos.
 R = (- ,1) U {2} U (3, +)
3
4
5
6
7
8
9
Note: there is no gap when
tracing the graph by a
vertical line from left to
right, then the set of real
numbers is the domain. For
the range, there is a gap
when tracing by horizontal
line from bottom going up
-2
Y=2x-1
2
-8
Example 2: Find the domain and range of the function defined
below
y
f(x)=1-x^2
f(x)=3x+1
1 − 𝑥2 𝑖𝑓 𝑥 < 0
8
f(x) =
Y=3x+1
6
4
3𝑥 + 1 𝑖𝑓 1 ≤ 𝑥
2
x
-7
-6
-5
-4
-3
-2
-1
1
-2
Y=1-x2
-4
-6
 D= (- ,0) U [1, +)
 R = (- ,1) U [4, +)
-8
2
3
4
5
6
7
Note: there is a gap (from 0
to 1) when tracing the
graph by a vertical line from
left to right. For the range,
there is a gap ( from 1 to 4)
when tracing by horizontal
line from bottom going up
4
8/24/2021
Example 3: Find the domain and range of the function below
f(x)=1-x
f(x) =
𝑥+1
𝑖𝑓 𝑥 − 2
4 − 𝑥2
𝑖𝑓 − 2 < 𝑥 < 2
1 − 𝑥 𝑖𝑓 2 ≤ 𝑥
Note: for domain, there is
no gap from left to right
when tracing the graph by a
vertical line. For the range,
there is a gap ( from -1 to 0)
when tracing by horizontal Y=x+1
line from bottom going up
 D= all real nos.
 R = (- ,-1] U (0, 2]
f(x)=x+1
f(x)=sqrt (4-x^2)
5
Y= 4 − 𝑥2
x
-5
Y=1-x
5