Question 112746
I think you mean: If {{{f(x)=sqrt(x-1)}}}, then find {{{f^(-1)(x)}}}.


The first thing you have to do is determine whether {{{f^(-1)(x)}}} exists.  There is a pretty simple test that will tell you.  It is called the horizontal line test.


Step 1: Graph the original function.


{{{graph(400,400,-2,5,-2,5,sqrt(x-1))}}}


Step 2:  If ANY possible horizontal line intersects the graph in more than one place, then we know that {{{f^(-1)(x)}}} is NOT a function, i.e. f does NOT have an inverse.  Otherwise, {{{f^(-1)(x)}}} does exist and f does have an inverse.  For this example, there is no horizontal line that intersects the graph more than once, so we know that f has an inverse.  Now, and only now, can we set about finding it.


Procedure to find the inverse of a one-to-one function (one that has an inverse):


Step 1:  Replace f(x) with y.


{{{y=sqrt(x-1)}}}


Step 2:  Swap the positions of the x and y variables with each other.


{{{x=sqrt(y-1)}}}


Step 3:  Rearrange the equation so that it again is showing y as a function of x, i.e. solve for y.


{{{x=sqrt(y-1)}}}
{{{x^2=y-1}}}
{{{y=x^2+1}}}



Ok, Super-Double-Plus Extra Credit.  If {{{f(x)=x^2+1}}}, does {{{f^(-1)(x)}}} exist?


Hope this helps
John