SOLUTION: If g(x)=1-x^2 and f(g(x))=(1-x^2)/x^2 when x does not equal 0, what is the value of f(1/2)?

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Question 959132: If g(x)=1-x^2 and f(g(x))=(1-x^2)/x^2 when x does not equal 0, what is the value of f(1/2)?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
One way to solve this is to find f(x) first and then use that for find f(1/2).
To find f(x) from f(g(x)) we look to express %281-x%5E2%29%2Fx%5E2 in terms of g(x). The numerator of f(g(x)) is already equal to g(x). For the denomoinator, x^2, we need to solve g%28x%29+=+1-x%5E2 for x%5E2. Adding x%5E2 to both sides we get:
x^2 + g(x) = 1
Subtracting g(x) from each side we get:
x^2 = 1 - g(x)

Now we express f(g(x)) in terms of g(x):
f%28g%28x%29%29+=+g%28x%29%2F%281-g%28x%29%29
From this we can see where the input to f goes. Using this pattern for f(x) we get:
f%28x%29+=+x%2F%281-x%29

Now we can find f(1/2):
f%281%2F2%29+=+%281%2F2%29%2F%281-%281%2F2%29%29
Simplifying...
f%281%2F2%29+=+%281%2F2%29%2F%281%2F2%29
f%281%2F2%29+=+1