document.write( "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)? \n" ); document.write( "

Algebra.Com's Answer #586280 by jsmallt9(3758) $\"\"$ $\"About$

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).\r
\n" ); document.write( "\n" ); document.write( "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:
\n" ); document.write( "x^2 + g(x) = 1
\n" ); document.write( "Subtracting g(x) from each side we get:
\n" ); document.write( "x^2 = 1 - g(x)

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

\n" ); document.write( "Now we can find f(1/2):
\n" ); document.write( " $\"f%281%2F2%29+=+%281%2F2%29%2F%281-%281%2F2%29%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"f%281%2F2%29+=+%281%2F2%29%2F%281%2F2%29\"$
\n" ); document.write( " $\"f%281%2F2%29+=+1\"$ \n" ); document.write( "

\n" );