SOLUTION: if f(x) = ax+b/cx+d, x≠ -d/c-a/c, find f^-1(x)

Algebra ->  Functions -> SOLUTION: if f(x) = ax+b/cx+d, x≠ -d/c-a/c, find f^-1(x)      Log On


   

Question 1179409: if f(x) = ax+b/cx+d, x≠ -d/c-a/c, find f^-1(x)
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29+=+%28ax%2Bb%29%2F%28cx%2Bd%29

Write y for f(x)

y+=+%28ax%2Bb%29%2F%28cx%2Bd%29

Interchange x and y

x+=+%28ay%2Bb%29%2F%28cy%2Bd%29

Solve for y:

x%28cy%2Bd%29+=+ay%2Bb

cxy%2Bdx+=+ay%2Bb

cxy-ay+=+b+-+dx

y%28cx-a%29+=+%28b-dx%29

y+=+%28b-dx%29%2F%28cx-a%29 

Write f-1(x) for y:

f-1(x) = %28b-dx%29%2F%28cx-a%29 

Edwin

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The other tutor showed a derivation for the answer:

The inverse of %28ax%2Bb%29%2F%28cx%2Bd%29 is %28b-dx%29%2F%28cx-a%29

This particular kind of inverse is encountered often, especially in timed math competitions. While the derivation is not very complicated, it takes time and effort. It might be worthwhile to memorize the answer.

Write the inverse in a slightly different form:

The inverse of %28ax%2Bb%29%2F%28cx%2Bd%29 is %28-dx%2Bb%29%2F%28cx-a%29

Note the pattern: The b and c stay where they are; the a and d switch places and switch signs.