document.write( "Question 1184854: Given that f(x) = 1 + 4x - x^2 for x ≥ 2. Find the coordinates of the turning point of the function f(x),stating whether it is a maximum or minimum point. Explain why f(x) has an inverse and find an expression for f^-1(x) in terms of x. \n" ); document.write( "

Algebra.Com's Answer #815501 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%22f%27%28x%29%22+=+4-2x\"$ ===> the turning point is at x = 2, since $\"%22f%27%282%29%22+=+0\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%22f%27%27%28x%29%22+=+-2+%3C+0\"$ , and in particular $\"%22f%27%27%282%29%22+=+-2+%3C+0\"$ ===> there is a maximum point at x = 2.
\n" ); document.write( "(This is to be expected since the coefficient of $\"x%5E2\"$ is negative, and so the parabola opens downward.)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "For $\"x%3E=2\"$ , f(x) has an inverse function because it is one-to-one over this set. The function is one-to-one because it is strictly decreasing over [2, $\"%2Binfinity\"$ ).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The inverse function is obtained as follows:\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+-x%5E2+%2B+4x+%2B+1\"$ ===> $\"0+=+-x%5E2+%2B+4x+%2B+%281-y%29\"$ ===> $\"x+=+2+%2B-+sqrt%285-y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "after solving x in terms of y using the quadratic formula. Choose $\"x+=+2+%2B+sqrt%285-y%29\"$ since $\"x%3E=2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Interchanging the places of x and y, the inverse function is therefore $\"f%5E%28-1%29%28x%29+=+2+%2B+sqrt%285-x%29\"$ .
\n" ); document.write( " \n" ); document.write( "

\n" );