

The principal square root function
 the inverse of the square of x
(or quadratic) function 
Definition
of the inverse function 
The
graph of the principal square root function 
Translation
of the principal square root function in the direction of the
xaxis 





The principal square root function
 the inverse of the square of x
(or quadratic) function 
Definition
of the inverse function 
The
inverse function is a function, usually written f
^{}^{1},
whose domain and range are respectively the range and
domain of a given function f,
that is 

f
^{}^{1}(x)
= y if
and only if f
(y)
= x, 


or it is
the function whose composition with the given function
is the identity function, i.e., 





In
order that the inverse should have a unique value for each
argument, and so be properly a function, the
extraction of positive square roots
is the inverse of squaring, since 

however,
without the restriction to positive values, the square root
function on the domain of real numbers does not have an inverse. 

The
graph of the principal square root function 
The
graph of the inverse function is the reflection about the line y =
x
of the graph of a given function. 
A
function f has an inverse if and only if when its graph is
reflected about the line y =
x, the result is the graph of a
function that passes the vertical line test. 
A
relation is a function if there are no vertical lines that
intersect the graph at more than one point. 


Translation of the principal square root function
in the direction of
the xaxis 
Horizontal
line test 
A
function f
has an inverse if no horizontal line intersects the graph of f
more than once. 
If
any horizontal line intersects the graph of f
more than once, then f
does not have an inverse. 
A
mapping associating a unique member of the codomain with every
member of the domain of a function is called one to one
correspondence. 
A
function f
is onetoone if and only if f
has an inverse. 
Given
f
(x)
= x^{2} + y_{0}_{
}and,
since f
[f
^{}^{1}(x)]
= x 
therefore, f
[f
^{}^{1}(x)]
= [f
^{}^{1}(x)]^{2
}+ x_{0}
= x, or
[f
^{}^{1}(x)]^{2 }
= x 
x_{0}

then 



To find the
yintercept,
set x
= 0 and solve for y, that is, 



The
graph of
translated principal square root function
in the direction of
the xaxis 









College
algebra contents 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 