SOLUTION: find the inverse of following functions: (a) f(x)=(x^3 + 5)/(x-3) , x≠3 (b)f(x)=(x^3 - 8)/(x^2 - 4) , x≠±2

Algebra ->  Functions -> SOLUTION: find the inverse of following functions: (a) f(x)=(x^3 + 5)/(x-3) , x≠3 (b)f(x)=(x^3 - 8)/(x^2 - 4) , x≠±2      Log On


   

Question 346009: find the inverse of following functions:
(a) f(x)=(x^3 + 5)/(x-3) , x≠3

(b)f(x)=(x^3 - 8)/(x^2 - 4) , x≠±2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To find inverses:
  1. Replace the function notation, if any, with a "y".
  2. Rewrite the equation writing an "x" where you have a "y" and writing a "y" where you have a "x". This step is where you are changing the function to its inverse.
  3. If possible, solve the resulting equation for y. If you are able to find a single expression for y, then the inverse of the function is a function, too.

Let's try this out:
f%28x%29=%28x%5E3+%2B+5%29%2F%28x-3%29
1) Replace function notation:
y+=+%28x%5E3+%2B+5%29%2F%28x-3%29
2) Swap the x's and y's:
x+=+%28y%5E3+%2B+5%29%2F%28y-3%29
This is the inverse relation.
3) Solve fo y, if possible.
Multiply both sides by y-3:
xy+-+3x+=+y%5E3+%2B+5
Gather the y terms on one side and everything else on the other side of the equation. I'm going to subtract xy and 5 from each side:
-3x+-5+=+y%5E3+-xy
At this point I don't see a way to solve for y. So the inverse is not a function.

f%28x%29=%28x%5E3+-+8%29%2F%28x%5E2+-+4%29
1) Replace function notation:
y+=+%28x%5E3+-+8%29%2F%28x%5E2+-+4%29
2) Swap the x's and y's:
x+=+%28y%5E3+-+8%29%2F%28y%5E2+-+4%29
This is the inverse relation.
3) Solve for y, if possible
Multiply both sides by y%5E2+-+4
xy%5E2+-+4x+=+y%5E3+-+8
Gather the y terms on one side:
-4x+%2B+8+=+y%5E3+-xy%5E2
And again I don't see a way to solve for y. So this inverse in not a finction either.