How do you find the inverse of #f(x)=(8x-4)/(2x+6)# and graph both f and #f^-1#?

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Answer 1 · 2017-09-21T13:01:47

# f^(-1)(x) = (6x+4)/(8-2x) #

We have:

# f(x) = (8x-4)/(2x+6) #

Whose graph is:
graph{(8x-4)/(2x+6) [-18, 18, -20, 20]}

To determine the inverse, #f^(-1)(x)#, we take the given equation, and rearrange in the form #x=g(f)# then #f^(-1)(x)=g(x)#

# f = (8x-4)/(2x+6) #

# :. (2x+6)f = 8x-4 #

# :. 2xf+6f = 8x-4 #

# :. 8x-2xf = 6f+4 #

# :. x(8-2f) = 6f+4 #

Leading to:

# x = (6f+4)/(8-2f) #

Hence:

# f^(-1)(x) = (6x+4)/(8-2x) #

Whose graph is:
graph{(6x+4)/(8-2x) [-18, 18, -20, 20]}

And, as expected, we note that both #f(x)# and #f^(-1)(x)# are reflections in the line #y=x#, as we see when display both on the same graph:

graph{(y- (8x-4)/(2x+6))(y - (6x+4)/(8-2x))(y-x)=0 [-18, 18, -20, 20]}