SOLUTION: Hello everyone, The question: Find the inverse of the function f(x) = {{{ sqrt (x^2 + (1/x^2)) }}} I am having trouble isolating the y for the inverse function. I would great

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hello everyone, The question: Find the inverse of the function f(x) = {{{ sqrt (x^2 + (1/x^2)) }}} I am having trouble isolating the y for the inverse function. I would great      Log On


   

Question 105421: Hello everyone,
The question: Find the inverse of the function f(x) = +sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+
I am having trouble isolating the y for the inverse function. I would greatly appreciate any help. My work is shown below.
Thank you very much!
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My work :
f(x) = +sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+
y = +sqrt+%28x%5E2+%2B+%281%2Fx%5E2%29%29+
x = +sqrt+%28y%5E2+%2B+%281%2Fy%5E2%29%29+
= +sqrt+%28%28y%5E4+%2B+1%29%2Fy%29+
+x%5E2+=+%28y%5E4+%2B+1%29%2Fy%5E2+
+x%5E2y%5E2+=+y%5E4+%2B+1+
+x%5E2y%5E2+-+y%5E4+=+1+
+y%5E2%28x%5E2+-+y%5E2%29+=+1+
Answer by TP(29) About Me  (Show Source):
You can put this solution on YOUR website!
Fine up to sqrt+%28y%5E2+%2B+1%2Fy%5E2%29.
So we'll go from there.
x=sqrt+%28y%5E2+%2B+1%2Fy%5E2%29
Square both sides:
x%5E2=+y%5E2+%2B+1%2Fy%5E2+=+y%5E4%2Fy%5E2+%2B+1%2Fy%5E2+=+%28y%5E4+%2B+1%29%2Fy%5E2.
Now multiply both sides by y%5E2
We get:
y%5E2x%5E2+=+y%5E4+%2B+1
Now subtract y%5E2x%5E2 from both sides:
We get y%5E4-x%5E2y%5E2+%2B+1+=+0 (i)
Let z+=+y%5E2 and (i) becomes:
z%5E2+-+x%5E2z+%2B+1+=+0
Now using the formula for the solution of az%5E2+%2B+bz+%2B+c+%2B+0 i.e.
z+=+%28-b%2B-sqrt%28b%5E2+-+4ac%29%29%2F2awhere a=1,+b=-x%5E2 and +c=1
we get:
z+=+%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2
But z+=+y%5E2 and so:
y%5E2+=+%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2
Hence y=+sqrt%28%28x%5E2%2B-sqrt%28x%5E4+-+4%29%29%2F2%29.
Now there are TWO possible answers for the solution to the equation but only ONE is correct for the inverse function and it is:
y+=+sqrt%28%28x%5E2+%2B+sqrt%28x%5E4+-+4%29%29%2F2%29ANS.
(NB. I'll leave it up to you to work out what y+=+sqrt%28%28x%5E2+-+sqrt%28x%5E4+-+4%29%29%2F2%29 represents.
You should also use your calculator to check the inverse function answer - I suggest, for example, that you replace x by 4 or a similar small value).