y = (2^(x+2) + 1) / 2^x
split the fraction into 2 separate denominators to get:
y = ((2^(x+2) / 2^x) + (1 / 2^x)
2^(x+2) / 2^x is equal to 2^(x+2 - x) which is equal to 2^2.
your equation becomes:
y = 2^2 + 1/2^x
simplify to get:
y = 4 + 1/2^x
solve for 1/2^x to get:
1/2^x = y - 4
solve for 2^x to get:
2^x = 1 / (y - 4)
replace x with y and y with x to get:
2^y = 1 / (x - 4)
take the log of both sides of the equation to get:
log(2^y) = log(1 / (x-4))
since log(2^y) is equal to y*log(2), the equation becomes:
y*log(2) = log(1 / (x-4))
divide both sides of the equation by log(2) to get:
y = log(1 / (x-4)) / log(2)
that's your inverse equation.
to prove that's it's an inverse equation, take the intersection of the line y = -x + c and the two equations.
the intersection on one of the equations should be (x,y).
the intersection on the other equation should be (y,x)
I used c = 20 in the following graph of both equations.
the equation i used was therefore y = -x + 20.
i also used y = -x + 10, but i'm only showing you the intersections of the two equations with the line y = -x + 20.
inverse equations are reflections about the line y = x.
the line y = -x is perpendicular to the line y = x.
the distance from either equation to the line y = x is the intersection of both equations with the line y = -x.
that's what the graph is showing.
here's the graph:
look below the graph for further comments.
you can see on the graph that the intersection of both equations with the line y = -x yields the point (-4.35,24.35) on the original equation, and the point (24.35, -4.35) on the inverse equation.
this confirms the equations are inverses of each other.