document.write( "Question 1144347: please help me solve this: if f(x)=x^2+2x+2,find two functions g for which (f°g)(x)=x^2-4x+5. \n" ); document.write( "

Algebra.Com's Answer #765488 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( " $\"f%28x%29+=+x%5E2%2B2x%2B2+=+%28x%5E2%2B2x%2B1%29%2B1+=+%28x%2B1%29%5E2%2B1\"$

\n" ); document.write( "

\n" ); document.write( "Therefore

\n" ); document.write( " $\"g%28x%29+=+x-3\"$
\n" ); document.write( " $\"f%28x%29+=+%28x%2B1%29%5E2%2B1\"$

\n" ); document.write( "It takes some experience with this kind of problem to see how the above process works.

\n" ); document.write( "Here is another way to solve the problem.

\n" ); document.write( "The given function f(x) contains a term in x^2; that means somewhere along the way the function f has to square the input. So as above we can write f(x) as

\n" ); document.write( " $\"f%28x%29+=+x%5E2%2B2x%2B2+=+%28x%5E2%2B2x%2B1%29%2B1+=+%28x%2B1%29%5E2%2B1\"$

\n" ); document.write( "Then since f(x) is quadratic and f(g(x)) is also quadratic, we know that g(x) must be linear.

\n" ); document.write( "So let g(x) = ax+b. Then

\n" ); document.write( "

\n" ); document.write( "Then since f(g(x)) = x^2-4x+5, equating coefficients gives us

\n" ); document.write( "(1) $\"a%5E2=1\"$
\n" ); document.write( "(2) $\"2ab%2B2a+=+-4\"$
\n" ); document.write( "(3) $\"b%5E2%2B2b%2B2+=+5\"$

\n" ); document.write( "(1) gives us a=1

\n" ); document.write( "Substituting a=1 in (2) gives us
\n" ); document.write( " $\"2b%2B2+=+-4\"$
\n" ); document.write( " $\"2b+=+-6\"$
\n" ); document.write( " $\"b+=+-3\"$

\n" ); document.write( "And now we know the linear function g(x) is ax+b = x-3.

\n" ); document.write( "ANSWER:
\n" ); document.write( " $\"f%28x%29+=+%28x%2B1%29%5E2%2B1\"$
\n" ); document.write( " $\"g%28x%29+=+x-3\"$ \n" ); document.write( "

\n" );