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JIACHENG CAI 1
MATH 124 NOTE 14 Section 5.1
5.1 Inverse Functions
1. One-to-One Functions
Example: Determine whether each of the following is one-to-one function.
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Example: Is each of the following function one-to-one?
(a) 𝑓(𝑥) = 3𝑥 + 4 (b) 𝑓(𝑥) = 𝑥2
2. The Inverse of a Function
Idea:
Domain of 𝑓−1=
Range of 𝑓−1=
Example: Show that 𝑓(𝑥) = 𝑥3 and 𝑔(𝑥) = 𝑥1/3 are inverses of each other.
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Example: Find the inverse of 𝑓(𝑥) =𝑥5−3
2
Example: Find the inverse of 𝑓(𝑥) =2𝑥+3
𝑥−1 , and state the range of the
inverse function.
Example: Find the inverse of 𝑓(𝑥) =5𝑥−3
−7𝑥+2 , and state the range of the
inverse function.
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3. Graphing the Inverse of a Function
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5.2 Exponential Functions and Graphs
Real life Example: Growth
4. Exponential Functions
Example: Graph 𝑓(𝑥) = 2𝑥
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Example: Graph 𝑓(𝑥) = (1
2)
𝑥
Example: Graph each of the following. Describe how each graph can be
obtained from the graph of 𝑓(𝑥) = 2𝑥
(a) 𝑓(𝑥) = 2𝑥−2 (b) 𝑓(𝑥) = 2𝑥 − 4 (c) 𝑓(𝑥) = 5 − 0.5𝑥
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2. Application
Example 4 (Compound Interest) The amount of money 𝐴 to which a
principle 𝑃 will grow after 𝑡 years at interest rate 𝑟, compounded 𝑛 times
per year, is given by the formula
𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛𝑡
Suppose that $100,000 is invested at 6.5% interest, compounded
semiannually.
(a) Find a function for the amount to which the investment grows after 𝑡
years.
(b) Find the amount of money in the account at 𝑡 = 0,4,8,10 years.
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3. The Number 𝑒
Idea: In the compound interest formula 𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛𝑡, Suppose $1 is
invested at 100% interest rate for 1 year. Then
𝐴 =
The Euler Number 𝑒 =
Graph of 𝑓(𝑥) = 𝑒𝑥 and 𝑔(𝑥) = 𝑒−𝑥