MATH1110 Lecture Notes - Lecture 2: Codex Corbeiensis I, Inverse Function, Inverse Trigonometric Functions
18 Jun 2018
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MATH1110: Mathematics 1
More Functions (II): Inverse Functions
1 Inverse Functions
Informally, given a function fthe inverse function, denoted by f−1, undoes
the original function. That is, if we start with any value in the domain of f,
xsay, and find f(x) then when we put the result into f1we will end up with
the original value x.
Example 1. For the function f(x)=2x+ 3 its inverse function is
f−1(x) = 1
2(x−3).
Let x= 2. To find f(2) we first multiply by 2 then add 3, i.e. f(2) = 7.
To undo this we should first subtract 3 and then divide by 2, i.e. f−1(7) =
7−3
2= 2, i.e. the value we started with.
It is somewhat unfortunate that the symbol for the inverse function is
f−1because here the 1 isnt to be interpreted as an exponent. That is
f−1(x)6=1
f(x).
We represent the reciprocal of the number f(x) by 1
f(x)=f(x)−1
Definition 1. Formally, for the function fits inverse function, denoted
by f−1is the function such that:
(f−1◦f)(x) = x∀x∈dom f.
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Document Summary
Informally, given a function f the inverse function, denoted by f 1, undoes the original function. That is, if we start with any value in the domain of f , x say, and nd f (x) then when we put the result into f 1 we will end up with the original value x. For the function f (x) = 2x + 3 its inverse function is f 1(x) = To nd f (2) we rst multiply by 2 then add 3, i. e. f (2) = 7. To undo this we should rst subtract 3 and then divide by 2, i. e. f 1(7) = = 2, i. e. the value we started with. It is somewhat unfortunate that the symbol for the inverse function is f 1 because here the 1 isnt to be interpreted as an exponent. We represent the reciprocal of the number f (x) by.
