Question 64824
FOR THE FUNCTION DEFINED BY {{{f(x)=2-x^2}}},{{{0<x}}}, USE A SKETCH TO HELP FIND A FORMULA FOR f^-1(x).
{{{f(x)=2-x^2}}} Is graphed below.  Ignore the left side of the graph because the domain includes only positive values.  I have also graphed the line y=x, because inverse functions are reflected about that line.
{{{graph(300,200,-10,10,-10,10,2-x^2,x)}}}
Notice that the graph has a y-intercept of (0,2), therefore the inverse will have an x-intercept of (2,0).  Notice also that the graph goes toward -infinity as x goes towards +infinity, therefore its inverse will have x's that go to -infinity as y goes to positive infinity.
The parent f(x)=x^2, therefore the parent f^-1(x)=sqrt(x), it has a horizontal reflection so f^-1(x)=sqrt(-x).  It also has a horizontal shift right 2 units, so the inverse function is:
f^-1(x)={{{sqrt(2-x)}}}
The graph of the inverse is below:
{{{graph(300,200,-10,10,-10,10,sqrt(2-x))}}}
If you aren't being taught about reflections and shifts, plot the points for f(x) and then reverse the x's and y's and you'll have the inverse.
Happy Calculating!!!