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

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