Extremum, Symmetry, Piecewise Functions, and
the Difference Quotient
Joseph Lee
Metropolitan Community College
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Increasing Functions, Decreasing Functions, Constant
Functions.
Let f be a function and (a, b) be some interval in the domain of f .
The function is called
increasing over (a, b) if f (x) < f (y ) for every x < y ,
decreasing over (a, b) if f (x) > f (y ) for every x < y , and
constant over (a, b) if f (x) = f (y ) for every x and y
(where a < x < y < b).
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1.
Increasing:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1.
Increasing: (−5, −2) ∪ (1, 4)
Decreasing:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1.
Increasing: (−5, −2) ∪ (1, 4)
Decreasing: (−1, 1) ∪ (4, 5)
Constant:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1.
Increasing: (−5, −2) ∪ (1, 4)
Decreasing: (−1, 1) ∪ (4, 5)
Constant: (−2, −1)
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1. (Continued)
Relative Maximum:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1. (Continued)
Relative Maximum: (4, 2)
Joseph Lee
Relative Minimum:
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1. (Continued)
Relative Maximum: (4, 2)
Relative Minimum: (1, −1)
Domain:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1. (Continued)
Relative Minimum: (1, −1)
Relative Maximum: (4, 2)
Domain: [−5, 5]
Range:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1. (Continued)
Relative Minimum: (1, −1)
Relative Maximum: (4, 2)
Domain: [−5, 5]
Range: [−2, 2]
Zeros of the function:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 1. (Continued)
Relative Minimum: (1, −1)
Relative Maximum: (4, 2)
Domain: [−5, 5]
Range: [−2, 2]
Zeros of the function: −3, 0, 2, 5
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2.
Increasing:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2.
Increasing: (−2, 0)
Decreasing:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2.
Increasing: (−2, 0)
Decreasing: (−3, −2) ∪ (0, 2)
Constant:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2.
Increasing: (−2, 0)
Decreasing: (−3, −2) ∪ (0, 2)
Constant: (−5, −3) ∪ (2, 5)
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2. (Continued)
Relative Maximum:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2. (Continued)
Relative Maximum: (1, −1)
Relative Minimum:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2. (Continued)
Relative Maximum: (1, −1)
Relative Minimum: (0, 2)
Domain:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2. (Continued)
Relative Maximum: (1, −1)
Relative Minimum: (0, 2)
Domain: (−5, 5)
Joseph Lee
Range:
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 2. (Continued)
Relative Maximum: (1, −1)
Relative Minimum: (0, 2)
Domain: (−5, 5)
Joseph Lee
Range: [−2, 2]
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 3.
Increasing:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 3.
Increasing: (−5, −1)
Decreasing:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 3.
Increasing: (−5, −1)
Decreasing: (−1, 0) ∪ (1, 5)
Constant:
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 3.
Increasing: (−5, −1)
Decreasing: (−1, 0) ∪ (1, 5)
Constant: (0, 1)
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Even and Odd Functions
A function f is called even if
f (−x) = f (x).
A function f is called odd if
f (−x) = −f (x).
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 4.
Determine if f is even, odd, or neither.
f (x) = x 2 − 4
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 4.
Determine if f is even, odd, or neither.
f (x) = x 2 − 4
f (−x) = (−x)2 − 4
= x2 − 4
= f (x)
Thus, f is an even function.
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 5.
Determine if g is even, odd, or neither.
g (x) = x 3 − 2x
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 5.
Determine if g is even, odd, or neither.
g (x) = x 3 − 2x
g (−x) = (−x)3 − 2(−x)
= −x 3 + 2x
= −(x 3 − 2x)
= −g (x)
Thus, g is an odd function.
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 6.
Determine if h is even, odd, or neither.
h(x) = (x − 2)2
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 6.
Determine if h is even, odd, or neither.
h(x) = (x − 2)2
h(−x) = [(−x) − 2]2
= x 2 + 4x + 4
Note:
h(x) = (x − 2)2 = x 2 − 4x + 4
−h(x) = −(x − 2)2 = −x 2 + 4x − 4
Thus, h is neither even nor odd.
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 7.
Evaluate the piecewise function.


2x + 8
f (x) = x 2


1
if x ≤ −2
if − 2 < x ≤ 1
if x > 1
f (−3) =
f (−1) =
f (2) =
f (4) =
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 8.
Evaluate the piecewise function.
(
x
f (x) =
−x
if x ≥ 0
if x < 0
f (−2) =
f (−1) =
f (1) =
f (2) =
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 9.
Graph the piecewise function.
(
x +2
f (x) =
1
if x ≤ 0
if x > 0
y
3
2
1
−3
−2
−1
1
2
3
x
−1
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 9.
Graph the piecewise function.
(
x +2
f (x) =
1
if x ≤ 0
if x > 0
y
3
2
1
−3
−2
−1
1
2
3
x
−1
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 10.
Graph the piecewise function.
(
x
f (x) =
−x
if x ≥ 0
if x < 0
y
3
2
1
−3
−2
−1
1
2
3
x
−1
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 10.
Graph the piecewise function.
(
x
f (x) =
−x
if x ≥ 0
if x < 0
y
3
2
1
−3
−2
−1
1
2
3
x
−1
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 11.
Graph the piecewise function.


2x + 8
f (x) = x 2


1
if x ≤ −2
if − 2 < x ≤ 1
if x > 1
y
4
3
2
1
−3 −2 −1
1
2
3
x
−1
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 11.
Graph the piecewise function.


2x + 8
f (x) = x 2


1
if x ≤ −2
if − 2 < x ≤ 1
if x > 1
y
4
3
2
1
−3 −2 −1
1
2
3
x
−1
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Difference Quotient.
For a function f (x) and an increment h, the difference quotient is
f (x + h) − f (x)
,
h
Joseph Lee
h 6= 0.
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 12.
Let f (x) = 2x + 3. Find the difference quotient.
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 12.
Let f (x) = 2x + 3. Find the difference quotient.
f (x + h) − f (x)
[2(x + h) + 3] − (2x + 3)
=
h
h
=
2x + 2h + 3 − 2x − 3
h
=
2h
h
(h 6= 0)
=2
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 13.
Let f (x) = 5x − 6. Find the difference quotient.
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 13.
Let f (x) = 5x − 6. Find the difference quotient.
f (x + h) − f (x)
[5(x + h) − 6] − (5x − 6)
=
h
h
=
5x + 5h − 6 − 5x + 6
h
=
5h
h
(h 6= 0)
=5
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 14.
Let f (x) = x 2 + 1. Find the difference quotient.
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 14.
Let f (x) = x 2 + 1. Find the difference quotient.
f (x + h) − f (x)
[(x + h)2 + 1] − (x 2 + 1)
=
h
h
=
x 2 + 2xh + h2 + 1 − x 2 − 1
h
=
2xh + h2
h
=
h(2x + h)
h
(h 6= 0)
= 2x + h
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 15.
Let f (x) = x 2 − 4x. Find the difference quotient.
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q
Example 15.
Let f (x) = x 2 − 4x. Find the difference quotient.
f (x + h) − f (x)
[(x + h)2 − 4(x + h)] − (x 2 − 4x)
=
h
h
=
x 2 + 2xh + h2 − 4x − 4h − x 2 + 4x
h
=
2xh + h2 − 4h
h
=
h(2x + h − 4)
h
(h 6= 0)
= 2x + h − 4
Joseph Lee
Extremum, Symmetry, Piecewise Functions, and the Difference Q