Functions

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Then g is onto, and f is not.

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Slide 8

Exponential Functions

The exponential function with base b

is the following function from R to R+ :

expb(x) = bx

b0=1 b-x = 1/bx

bubv = bu+v

(bu)v = buv

(bc)u = bucu

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Slide 9

Logarithic Functions

The logarithic function with base b

(b>0, b1)

is the following function from R+ to R:

logb(x) = the exponent to which b must raised to obtain x .

Symbolically, logbx = y  by = x .

Properties:

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Slide 10

One-to-one Correspondences

Definition: A one-to-one correspondence

(or bijection) from a set X to a set Y

is a function f:X→Y

that is both one-to-one and onto.

Examples:

1) Linear functions: f(x)=ax+b when a0

(with domain and co-domain R)

2) Exponential functions: f(x)=bx (b>0, b1)

(with domain R and co-domain R+)

3) Logarithic functions: f(x)=logbx (b>0, b1)

(with domain R+ and co-domain R)

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Slide 11

Inverse Functions

Theorem:

Suppose F: X→Y is a one-to-one correspondence.

Then there is a function F-1: Y→X defined as follows:

Given any element in Y,

F-1(y) = the unique element x in X

such that F(x)=y .

The function F-1 is called the inverse function for F.

Example:

The logarithic function with base b (b>0, b 1)

is the inverse of the exponential function with base b.

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