Question 110971
first recall the {{{definition}}} of inverse: 

A function {{{f}}} with domain {{{D}}} is said to be {{{one_to_one}}} if {{{NO}}} 

distinct points in {{{D}}} have same {{{image}}} under {{{f}}}, that is:

{{{f(x1) }}} {{{is_ not_ equal}}} to {{{f(x2)}}} whenever {{{x1}}} {{{is_ not_ 

equal}}}  to {{{x2}}}, and ({{{x1}}},{{{x2}}}) is element of {{{D}}}.

 Why a polynomial function of even degree cannot have an inverse? 

Simply, because a polynomial function of even degree is {{{NOT}}} 

{{{one_to_one}}} function.

Each value of {{{x}}} squared, raised to {{{4th}}} degree, or higher even 

degree, will be that same value; for example, {{{-2^2= 4}}}, and also {{{2^2 = 

4}}}. This means that two distinct points in {{{D}}} have same {{{image}}} 

under {{{f}}}. 


example:

let {{{f(x) = 1 - x^2}}}

let domain {{{D}}} be equal to {{{-3}}},{{{-1}}},{{{0}}},{{{1}}},{{{3}}}

find {{{f(-3)}}}, {{{f(-1)}}},{{{f(0)}}},{{{f(1)}}},{{{f(3)}}}


    {{{f(-3)= 1 - (-3)^2= 1 - 9 = -8}}}

    {{{f(-1)= 1 - (-1)^2= 1 - 1 = 0}}}
    
{{{f(0)= 1 - (0)^2= 1 - 0 = 1}}} 

    {{{f(1)= 1 - (1)^2= 1 - 1 = 0}}}

    {{{f(3)= 1 - (3)^2= 1 - 9 = -8}}}

as you can see,  {{{f(-3)}}} and {{{f(3)}}} have same image under {{{f}}}

also {{{f(-1)}}} and {{{f(1)}}} have same image under {{{f}}}