document.write( "Question 1028867: Find the inverse function of f:[3,∞) -> R, f(x) = (x-4)^2 +1
\n" ); document.write( "Why does the domain of f need to be restricted to [4,∞) in this question? \n" ); document.write( "

Algebra.Com's Answer #643924 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here is the basic outline for finding the inverse
\n" ); document.write( "A) Replace f(x) with y (step 2 in table below)
\n" ); document.write( "B) Swap x and y (step 3 in table below)
\n" ); document.write( "C) Solve for y (steps 4 through 7 in table below)\r
\n" ); document.write( "\n" ); document.write( "Let's use this to find the inverse of the given function\r
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Number	Statement	Reason/Explanation
1.	$\"f%28x%29+=+%28x-4%29%5E2+%2B1\"$	None needed. This is the original function given
2.	$\"y+=+%28x-4%29%5E2+%2B1\"$	Replace f(x) with y
3.	$\"x+=+%28y-4%29%5E2+%2B1\"$	Swap x and y. Now we isolate y.
4.	$\"x-1+=+%28y-4%29%5E2\"$	Subtract 1 from both sides
5.	$\"%28y-4%29%5E2+=+x-1\"$	Flip the equation
6.	$\"y-4+=+sqrt%28x-1%29\"$	Take the square root of both sides. See note below
7.	$\"y=+sqrt%28x-1%29%2B4+\"$	Add 4 to both sides

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\n" ); document.write( "\n" ); document.write( "Note: the domain of [4,infinity) for f(x) turns out to be the range of the inverse function. Domain of original = range of inverse. In order to stretch onto positive infinity, we need to use the plus version of the plus/minus. So instead of using plus/minus, we can just use plus all by itself. So instead of using $\"y-4+=+%22%22%2B-sqrt%28x-1%29\"$ we stick with $\"y-4+=+sqrt%28x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "The inverse function is therefore

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\n" ); document.write( "\n" ); document.write( "The domain of f(x) must be restricted to [4,infinity) because including 3 in the domain makes the function not one-to-one. Notice how f(3) = f(5) = 2. You must restrict the domain to make f(x) one-to-one in order for f(x) to be invertible to a function. \n" ); document.write( "

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