## Tuesday, November 27, 2007

### Inverse function

Example 1: Find the inverse function of f given by

f(x) = 2x + 3

Solution to example 1:

write the function as an equation.

y = 2x + 3

solve for x.

x = (y - 3)/2

now write f-1(y) as follows .

f -1(y) = (y - 3)/2

or

or f -1(x) = (x - 3)/2

Check

f(f -1(x))=2(f -1(x)) + 3

=2((x-3)/2)+3

=(x-3)+3

=x

f -1(f(x))=f -1(2x+3)

=((2x+3)-3)/2

=2x/2

=x

conclusion: The inverse of function f given above is f -1(x) = (x - 3)/2

The properties of inverse functions are listed and discussed below.

Only one to one functions have inverses

If g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other.

If f and g are inverses of each other then both are one to one functions.

f and g are inverses of each other if and only if

(f o g)(x) = x , x in the domain of g

and

(g o f)(x) = x , x in the domain of f

Example

Let f(x) = 3 x and g(x) = x / 3

(f o g)(x) = f( g(x) ) = 3 ( x / 3 ) = x

and g o f)(x) = g( f(x) ) = (3 x) / 3 = x

Therefore f and g given above are inverses of each other.

If f and g are inverses of each other then

the domain of f is equal to the range of g

and

the range of f is equal to the domain of g.

Example

Let f(x) = sqrt (x - 3)

The domain of f is given by the interval [3 , + infinity)

The range of f is given by the interval [0, + infinity)

Let us find the inverse function

Square both sides of y = sqrt (x - 3) and interchange x and y to obtain the inverse

f -1 (x) = x 2 + 3

According to property 5,

The domain of f -1 is given by the interval [0 , + infinity)

The range of f -1 is given by the interval [3, + infinity

http://www.analyzemath.com/inversefunction/Tutorials.html

f(x) = 2x + 3

Solution to example 1:

write the function as an equation.

y = 2x + 3

solve for x.

x = (y - 3)/2

now write f-1(y) as follows .

f -1(y) = (y - 3)/2

or

or f -1(x) = (x - 3)/2

Check

f(f -1(x))=2(f -1(x)) + 3

=2((x-3)/2)+3

=(x-3)+3

=x

f -1(f(x))=f -1(2x+3)

=((2x+3)-3)/2

=2x/2

=x

conclusion: The inverse of function f given above is f -1(x) = (x - 3)/2

The properties of inverse functions are listed and discussed below.

Only one to one functions have inverses

If g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other.

If f and g are inverses of each other then both are one to one functions.

f and g are inverses of each other if and only if

(f o g)(x) = x , x in the domain of g

and

(g o f)(x) = x , x in the domain of f

Example

Let f(x) = 3 x and g(x) = x / 3

(f o g)(x) = f( g(x) ) = 3 ( x / 3 ) = x

and g o f)(x) = g( f(x) ) = (3 x) / 3 = x

Therefore f and g given above are inverses of each other.

If f and g are inverses of each other then

the domain of f is equal to the range of g

and

the range of f is equal to the domain of g.

Example

Let f(x) = sqrt (x - 3)

The domain of f is given by the interval [3 , + infinity)

The range of f is given by the interval [0, + infinity)

Let us find the inverse function

Square both sides of y = sqrt (x - 3) and interchange x and y to obtain the inverse

f -1 (x) = x 2 + 3

According to property 5,

The domain of f -1 is given by the interval [0 , + infinity)

The range of f -1 is given by the interval [3, + infinity

http://www.analyzemath.com/inversefunction/Tutorials.html