Given:
Graph the function #f(x)=(x+1)^2-4# and its inverse#color(blue)(f^-1(x)#.
A function and it's inverse will be symmetric around the line #color(red)(y=x#
Switch the position of #color(red)(x# and #color(red)(y# variables to find the inverse of a function.
We have,
#y=f(x)=(x+1)^2-4#
Write the quadratic as #color(blue)(x=(y+1)^2 - 4#, after switching #color(red)(x# and #color(red)(y# positions.
Next, solve #color(blue)(x=(y+1)^2-4#, for #color(red)(y#.
Switch sides and rewrite as
#color(blue)((y+1)^2-4=x#
Ad #color(red)(4# to both sides.
#(y+1)^2 - 4 + 4 = x+4#
#i.e., (y+1)^2 - cancel(4) + cancel(4) = x+4#
#i.e., (y+1)^2 =x+4#
#i.e., (y+1) =+-sqrt(x+4)#
Subtract #color(red)(1# from both the sides of the equation.
#i.e., y+1 - 1 =+-sqrt(x+4)-1#
#i.e., y+cancel(1) - cancel(1) =+-sqrt(x+4)-1#
Hence, we get
#y =f^-1(x)=+-sqrt(x+4)-1#
Our final solutions to the quadratic equation are
#y_1 = sqrt(x+4)-1# and
#y_2 = -sqrt(x+4)-1#
Examine the image of the graph containing #color(red)(f(x)# and #color(red)(f^-1(x)# below:
A graphical display calculator may also be used to draw the graph as shown below:
Hope it helps.
