Section 1.8 Conclusion
Inverse Functions
Definition (p. 37)
A function 𝑔 is the inverse function of the function 𝑓 if
Inverse
aallFunction
𝑓 𝑔 𝑥 of
= 𝑥 for
𝑥 in the domain of 𝑔
and
𝑔 𝑓 𝑥
= 𝑥 for all 𝑥 in the domain of 𝑓.
The function 𝑔 is denoted by 𝑓 −1 (read “𝑓 inverse”).
1
1
f ( f ( x))  f ( f ( x))  x
Note
𝑓 −1 does NOT mean “𝑓 raised to the power of negative one.”
Horizontal Line Test
Example 1
Sketch the
inverse of the
function
1
𝑓 𝑥 =
𝑥−3
Example 2
Sketch the
graph of the
inverse of the
function.
Section 3.1
Exponential Functions
Example 1
Approximate using a calculator.
a. 4
8
b. 𝑒 −4
Exponential Functions
Example 2
Graph the function
by making a table.
y
5
𝑓 𝑥 = 2𝑥
𝒙
𝒚
x
-5
-5
5
Example 3
Graph the function by making a table.
2
𝑓 𝑥 =
3
𝒙
𝑥
y
5
𝒚
x
-5
-5
5
Example 4
Use transformation of 𝑦 = 2𝑥 to graph the function.
y
𝑓 𝑥 = 2𝑥+2 − 1
5
Asymptote(s):
x
Domain:
Range:
-5
-5
5
Example 5
Use transformation of 𝑦 = 2𝑥 to graph the function.
y
𝑓 𝑥 = −2𝑥
5
Asymptote(s):
x
Domain:
Range:
-5
-5
5
Example 6
Use transformation of 𝑦 = 2𝑥 to graph the function.
y
𝑓 𝑥 = 2−𝑥
5
Asymptote(s):
x
Domain:
Range:
-5
-5
5
The Natural Base 𝒆
• Irrational number
(transcendental actually . . . like 𝜋)
• 𝑒 is called the “natural base” and 𝑓 𝑥 = 𝑒 𝑥 is
called the “natural exponential function.”
• 𝑒 ≈ 2.718281827
• Formal Definition of 𝑒:
1
lim 1 +
𝑛→∞
𝑛
𝑛
Example 7
Use transformation of 𝑦 = 𝑒 𝑥 to graph the function.
y
1 𝑥−1
𝑓 𝑥 = 𝑒
2
5
Asymptote(s):
x
Domain:
Range:
-5
-5
5
Example 8
Use transformation of 𝑦 = 𝑒 𝑥 to graph the function.
y
𝑓 𝑥 = −𝑒 4𝑥
5
Asymptote(s):
x
Domain:
Range:
-5
-5
5
Applications of Exponential Functions
• Used to model many real-life situations
▫ Financial Situations
▫ Population Growth
Compound Interest
• Note:
“Semiannually” means 𝑛 = 2
“Quarterly” means 𝑛 = 4
“Monthly” means 𝑛 = 12
Example 9
Suppose you have $12,000 to invest. Which
investment yields the greater return over 3 years:
7% compounded monthly or 6.85% compounded
continuously?
Example 10
The exponential function 𝑓 𝑥 = 574(1.026)𝑥
models the population of India (in millions) 𝑥
years after 1974.
a. What was India’s population to the nearest
million in 1974?
b. Find India’s predicted population to the
nearest million in 2028.
Example 11
Give the equation of
the exponential
function shown.
Questions???
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