Lesson 1.3
More on Functions
An Important Calculus Concept
 Determining
where (or if) a function is
increasing, decreasing or constant

Heavily relies on the “Difference Quotient.”
Difference Quotient Formula
f ( x  h)  f ( x )
h
Graphically . . .
Example 1(a)
Given that 𝑓 𝑥 = −2𝑥 2 + 𝑥 + 5, find
and simplify
𝑓 𝑥 + ℎ − 𝑓(𝑥)
.
ℎ
Example 1(b)
Given that 𝑓 𝑥 =
2
,
𝑥+2
find and simplify
𝑓 𝑥 + ℎ − 𝑓(𝑥)
.
ℎ
Increasing,
Decreasing or Constant
In Calculus, the difference quotient is used
to help you determine where the fnc. is . . .
Increasing:
𝒙𝟏 < 𝒙𝟐 implies 𝒇 𝒙𝟏 < 𝒇(𝒙𝟐 )
“the graph goes up from left to right”
Decreasing:
𝒙𝟏 < 𝒙𝟐 implies 𝒇 𝒙𝟏 > 𝒇(𝒙𝟐 )
“the graph goes down from left to right”
Constant:
𝒙𝟏 < 𝒙𝟐 implies 𝒇 𝒙𝟏 = 𝒇(𝒙𝟐 )
“the graph remains the same from left to right”
Increasing & Decreasing
Constant
Increasing & Decreasing
Example 2
Find the Intervals on the domain for which the
fnc. is increasing, decreasing or constant.
Relative Extrema
 Based
on the value of the fnc. (i.e. the 𝑦values)
Relative Maximum
Relative Minimum
 We
“peak” in the graph
“valley” in the graph
say . . .
“The function has a relative maximum of
(𝑦- value) at (𝑥-value).”
Relative Maxima and Minima
Relative Maxima and Minima
(cont.)
State the
extrema if
any.
Even Functions
 Symmetric
about the 𝑦axis.

𝑓 −𝑥 = 𝑓(𝑥)
for all 𝑥 in the
domain of 𝑓.
Odd Functions
 Symmetric
about
the origin.

𝑓 −𝑥 = −𝑓(𝑥)
for all 𝑥 in the
domain of 𝑓.
Testers
What type of symmetry? Is it even, odd, or
neither?
Testers
What type of symmetry? Is it even, odd, or
neither?
Testers
What type of symmetry? Is it even, odd, or
neither?
Testers
What type of symmetry? Is it even, odd, or
neither?
Example 3
Determine algebraically whether each fnc. is
even, odd, or neither.
a)
𝑓 𝑥 = 𝑥2 + 6
b)
𝑓 𝑥 = 7𝑥 3 − 𝑥
Example 3 (cont.)
Determine algebraically whether each fnc. is
even, odd, or neither.
a)
ℎ 𝑥 = 𝑥5 − 𝑥 + 1
b)
𝑓 𝑥 = 𝑥2 1 − 𝑥2
Piecewise Functions
Piecewise Function
A function that is defined differently for
different parts of the domain
“a function composed of different pieces”
Example 5
Evaluate.
𝑥 2 − 25
ℎ 𝑥 = 𝑥−5 ,
10,
a.
ℎ(3)
b.
ℎ(5)
𝑥≠5
𝑥=5
Graphing Piecewise Fnc.’s
−3,
2
−3
+
𝑥
,
𝑓 𝑥 =
𝑥
− 1,
2
𝑥≤0
0<𝑥≤2
𝑥>2
Is it a function?
What’s the range?
Example 6
Graph the piecewise fnc.
4,
𝑓 𝑥 = 𝑥 + 1,
−𝑥,
y
10
𝑥 ≤ −2
−2 < 𝑥 < 3
𝑥≥3
x
-10
-10
10
Questions???
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