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There are times when various quantities can be related to one another.
If this relationship follows a regular pattern then we can assign variables to these quantities
and write an equation to express this relationship.
This relationship is called a function.
A function can be as simple as a table of pairs of values.
The information about functions we require can be found in functions.
Here we will add the notions of composite functions and inverse functions.
First let's review the types of functions then talk about combinations of functions.
Finally we'll talk about inverse functions.
Functions
can be described by the way they map the independent variables to the
dependent variables. If two or more values map to the same value
we have many-to-one function.
Suppose we have the function y2 = x. For x = 4
y can equal 2 or -2. In this case one value for x
gives two values for y. This function is a one-to-many
function.
Now suppose we have y = 2x. In this case for every
value of x is is a unique value for y, that is twice x. This function is a
one-to-one function.
It is possible to have functions that
are many-to-many mappings, but many of these functions cannot be
described by one equation, instead they are usually described with
conditional set notation relating the dependent variable to the
independent variable.
Functions
can be combined arithmetically (as long as domain constraints are
named as necessary.) For example:
suppose f(x) = 3x + 2
and g(x) = x – 10
where the Domain and
Range are the Real Numbers
f(x)
+ g(x) = (3x + 2) + (x – 10) = 4x – 8
f(x)
– g(x) = (3x + 2) – (x – 10) = 2x + 12
f(x)
· g(x) = (3x + 2) · (x – 10) = 3x2
– 28x – 20
f(x)
/ g(x) = (3x + 2) / (x – 10)
In
#4 above, notice that if x = 10 we'd be dividing by 0, so our domain
constraint would be x ≠ 10.
An interesting operation with functions is called composition of
functions. This operation substitutes one function into another function.
Using
our previous functions f and g, let's substitute g into f. We get:
f( g(x) ) = 3( g(x) ) +
2
f( g(x) ) = 3( x –
10 ) + 2
f( g(x) ) = 3x – 30 + 2
f º g = 3x –
28
note
the last equation, “f of g of x” is written as f º
g
let's
do another:
g
º f = g( f(x) ) – 10
=
(3x + 2 ) – 10
=
3x – 8
Inverse functions are functions that map a range back to the domain of another function.
A function will take you from its range to its domain and its inverse function will return you
from that range back to that domain.
Mathematically we say that if the function f is a one-to-one function than its
inverse function f -1 is unique and defined on the
range of f and for all x in that range
f ( f -1(x) ) = x
The -1 is not an exponent, it means inverse.
Examples:
1) find the inverse of the function f(x) = x 2.
First step: set f ( f -1(x) ) = x
Now evaluate the left side ( f -1(x) )2 = x
Now take the square root of both sides: f -1(x) = ±x½
2) if f(x) = mx + b find f -1(x)
set f( f -1(x) ) = x
evaluate left side:
m( f -1(x) ) + b = x
solve for f -1(x):
m( f -1(x) ) = x - b
f -1(x) = (x - b) / m
3) if f(x) = x 5 + 1 find f -1(x)
set f( f -1(x) ) = x
evaluate left side:
( f -1(x) ) 5 + 1 = x
solve for f -1(x):
( f -1(x) ) 5 = x - 1
f -1(x) = (x - 1)1/5
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