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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