Question 381432
{{{y=3x^2-4x+5}}}
Interchange {{{x}}} and {{{y}}} and solve for {{{y}}}.
{{{x=3y^2-4y+5}}}
{{{x-5=3(y^2-(4/3)y)}}}
{{{(x-5)/3=(y^2-(4/3)y+4/9)-4/9}}}
{{{(x-5)/3+4/9=(y-2/3)^2}}}
{{{(3x-15)/9+4/9=(y-2/3)^2}}}
{{{(3x-15+4)/9=(y-2/3)^2}}}
{{{(3x-11)/9=(y-2/3)^2}}}
{{{y-2/3=0 +- sqrt(3x-11)/3}}}
{{{y=2/3 +- sqrt(3x-11)/3}}}
This new y is the inverse of the original function.
{{{highlight(f^(-1)(x)=2/3 +- sqrt(3x-11)/3)}}}
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{{{drawing(300,300,-10,10,-10,10,grid(1),graph(300,300,-10,10,-10,10,0,3x^2-4x+5,2/3+sqrt(3x-11)/3),graph(300,300,-10,10,-10,10,0,3x^2-4x+5,2/3-sqrt(3x-11)/3)))}}}