document.write( "Question 1187396: Question 3.7/12\r
\n" ); document.write( "\n" ); document.write( "Let f(x)=(x+1)2\r
\n" ); document.write( "\n" ); document.write( "Give the largest domain on which f is one-to-one and non-decreasing=
\n" ); document.write( "Give the range of f=
\n" ); document.write( "Find the inverse of f restricted to the domain above f-1(x)=
\n" ); document.write( "Give the domain of f-1 =
\n" ); document.write( "Give the range of f-1= \n" ); document.write( "

Algebra.Com's Answer #818726 by KMST(5328) $\"\"$ $\"About$

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The function $\"f%28x%29=%28x%2B1%29%5E2\"$ is a quadratic function, meaning a polynomial function of degree 2.
\n" ); document.write( "Those functions have graphs like this $\"graph%28110%2C100%2C2%2C7.5%2C1.5%2C6.5%2C%28x-5%29%5E2%2B2%29\"$ or the same curve upside down.
\n" ); document.write( "They are symmetrical with a vertex that is a maximum or minimum, separating a decreasing branch from an increasing branch.
\n" ); document.write( "You realize that for $\"x=-1\"$ , $\"x%2B1=0\"$ and $\"f%28-1%29=0%5E2=0\"$ , but for any other number $\"f%28x%29%3E0\"$ .
\n" ); document.write( "The graph decreases from any value of $\"x%3C-1\"$ , and increases for $\"x%3E=-1\"$ .
\n" ); document.write( "The largest domain on which f is one-to-one and non-decreasing is [-1,infinity), or $\"x%3E=-1\"$ .
\n" ); document.write( "The range is [0,infinity), because $\"f%28-1%29=0\"$ , and for $\"x%3E=-1\"$ it increases without bounds.
\n" ); document.write( "To find the inverse we swap $\"y\"$ and $\"x\"$ in the function defined as $\"y=%28x%2B1%29%5E2\"$ only when $\"x%3E=-1\"$ <--> $\"x%2B1%3E=0\"$ , and solve for $\"y\"$
\n" ); document.write( "we get $\"x=%28y%2B1%29%5E2\"$ for $\"y%2B1%3E=0\"$ , whose solution is $\"sqrt%28x%29=y%2B1\"$ <--> $\"y=sqrt%28x%29-1\"$ .
\n" ); document.write( "The inverse function is $\"f%5E-1\"$ $\"%28x%29=sqrt%28x%29-1\"$ for $\"X%3E=0\"$ .
\n" ); document.write( " $\"f%5E-1\"$ $\"%28x%29=sqrt%28x%29-1\"$ has domain [0,infinity) or $\"x%3E=0\"$ and range [-1,infinity) or $\"y%3E=-1\"$
\n" ); document.write( "The domain of an inverse function is the range of the domain-restricted function, and the range of the inverse is the restricted dominion. \n" ); document.write( "

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