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## Homework Statement

For exercises 49 and 50 let f(x) = (ax + b)/(cx + d)

50. Determine the constants a, b, c, d for which f = f

^{-1}

## Homework Equations

I found in question 49 when they asked to find f

^{-1}that:

f

^{-1}= (dx - b)/(a - cx)

This was also the answer at the back of the book but you are free to double check.

## The Attempt at a Solution

I made f(x) = f

^{-1}(x)

(ax + b)/(cx + d) = (dx - b)/(a - cx)

a

^{2}x - acx

^{2}+ ab - bcx = d

^{2}x - bcx - bd

From this i realized that finding the values for b and c would be manageable:

For b:

ab + bd = cdx

^{2}+ d

^{2}x + acx

^{2}- a

^{2}x

b ( a + d) = cdx

^{2}+ d

^{2}x + acx

^{2}- a

^{2}x

b = (cdx

^{2}+ d

^{2}x+ acx

^{2}- a

^{2}x) / (a + d)

For c:

-acx - cdx

^{2}= d

^{2}x - bd - a

^{2}x -ab

c (-ax - dx

^{2}) = d

^{2}x - bd - a

^{2}x -ab

c = (d

^{2}x - bd - a

^{2}x -ab)/(-ax - dx

^{2})

However i do not know how to find the constants a and d because you cannot just factor them out without leaving them in. Please help, Thank you.