MATHEMATICS  THEORY



This section introduces the concept of an inverse function
and its properties.






OnetoOne Function



Since only a onetoone function has an
inverse function, this section will first introduce the concept of a
onetoone function.
If a function has a unique function value for each element in the domain,
then this function is a onetoone function. This concept can be expressed
with mathematics as
f(x_{1}) ≠ f(x_{2}) whenever
x_{1}≠ x_{2}




onetoone Function f(x) = x^{3}


For example, the function f(x) = x^{3} is a onetoone
function because no two x values can have the same f(x) value as seen on
the diagram.




Not a onetoone Function
f(x) = x^{2} 

On the other hand, function f(x) = x^{2} is not a onetoone
function because when x equals either +a or a, the function value is a^{2} see
diagram. The method used to determine whether a function is a
onetoone function or not is called the Horizontal line test. 



Horizontal Line Test
Intersects Graph More Than Once
Click here to view movie (22.0 KB) 

The horizontal line test gives a geometric method to determine
whether a function is onetoone function. It states:
a function is a onetoone function if and only if no horizontal line intersects
its graph more than once. For
example, the function on the left is not a onetoone function. Actually,
all decreasing and increasing functions are onetoone function and only
onetoone functions can have inverse function. 


Inverse Function



The definition of inverse function states:
Suppose the domain and the range of a onetoone function f is R_{1} and
R_{2} respectively. Then its inverse function f^{ 1 }has
R_{2} and
R_{1} as its domain and range, and f^{ 1} is defined
by
f^{ 1}(y) = x <=> f(x) = y
for any y in R_{2}. Notice that in this definition, f^{ 1} is
not an exponent which means
f^{ 1}(x) ≠ 1/f(x)





Domain and Inverse Function
Click here to view movie (22.0 KB)
Symmetric f(x) and f^{ '}(x)
Click here to view movie (22.0 KB) 

In order to find the inverse function, the equation for
x needs to be solved in terms of y, and then switch x with y. To better
under this method, the process of finding the inverse function of f(x)
= 1  2/x^{2} when x > 0 is presented. First, rearranging
the equation in terms of x gives:
2/x^{2} = 1  y
x = (2/(1  y))^{1/2}
Then switching x with y gives:
y = (2/(1  x))^{1/2}
Therefore the inverse function is f^{ 1}(x) = (2/(1  x))^{1/2}.
Notice that the domain of this example is x > 0, this is because the
function is a onetoone function in this range. However, in the domain
of (∞, ∞), the function is not a onetoone function as
shown on the left, and thus there is no inverse function in this domain.
The inverse function can also be determined using a geometric method by
reflecting the graph of the function with respect to the line y = x. The
graphic on the left shows function f(x) = 1  2/x^{2} and its inverse
function f^{ 1}(x) = (2/(1  x))^{1/2} are symmetric with
respect to y = x in the range between 0 and +∞.
Various theorem related to inverse functions that are given in the following
paraghaphs, will help to determine the continuity and differentiability
of a function. 






Theorem About Continuous

Continuous Function
f(x) and f^{ '}(x) 

If a onetoone function is continuous
in its domain, then its inverse function is also continuous.
Some of the continuous function are differentiable, so do the inverse
function. The theorem related to the derivative of inverse functions
is introduced below. 


Theorem About Differentiable

Differentiable
Inverse Function Example^{} 

If a onetoone function is differentiable
and the derivative of its inverse function is not zero at a point, than
the
inverse function is differentiable at this point and
g^{ '}(a) = 1/f^{ '}(g(a))
in which g is the inverse function of function f.
In order to master this concept, find g ^{'}(8) in which g ^{'}(x)
is the inverse function of f(x) = x^{3}.
It can be calculated that
f(2) = 8
Therefore, switching x and y gives the value of the inverse function
which is
g(8) = 2
Since function f(x) is a onetoone increasing continuous function,
its derivative
is
f ^{'}(x) = 3x^{2}
Using the theorem about differentiable gives
g^{ '}(8) = 1/f^{ '}(g(8))
= 1/f^{ '}(2) = 1/(3(2)^{2}) = 1/12



