let y = f(x).
you get y = 1/(2x^3-9)
replace y with x and x with y and you get:
x = 1/(2y^3-9)
that's your inverse equation.
what's left is to solve for y if you can.
if not, go with it the way it is.
this one can be solved for y.
start with x = 1/(2y^3-9)
multiply both sides of the equation by (2y^3-9) and divide both sides of the equation by x.
you will get:
2y^3 - 9 = 1/x
add 9 to both sides of the equation to get:
2y^3 = 1/x + 9
since 1/x + 9 is the same as (1+9x)/x, the equation becomes:
2y^3 = (1+9x)/x
divide both sides of the equation by 2 to get:
y^3 = (1+9x)/(2x)
take the third root of both sides of the equation to get:
y = ((1+9x)/(2x))^(1/3)
that's your inverse equation after you have solved for y.
the grpah of the equation plus its inverse is shown below.
the graphs also includes the line y = x which they need to be a reflection of in order to be inverse equations.
the graph also includes lines y = -x + 5 and y = -x - 5 thst shows that the points are reflections about the line y = x.
that's because (x,y) on the original equation equals (y,x) on the inverse equation.
