document.write( "Question 1181302: Given the equation: f(x)= -2(x-1)^2+4, Determine the equation of its inverse. Restrict the domain of f(x) so that it’s inverse is a function. Then, state the new equation of its inverse. \n" ); document.write( "
Algebra.Com's Answer #811188 by greenestamps(13195)\"\" \"About 
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\n" ); document.write( "The given function is

\n" ); document.write( "\"y=-2%28x-1%29%5E2%2B4\"

\n" ); document.write( "This is a downward-opening parabola with vertex at (1,4). The function involves squaring an expression containing the variable; the inverse will involve taking the square root of an expression containing the variable. Since \"square root\" gives a positive result, the function needs to be restricted to a domain defined by (x-1) greater than or equal to 0. So the domain of the restricted function is [1,infinity).

\n" ); document.write( "Here is a graph of the (unrestricted) function:

\n" ); document.write( "\"graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C-2%28x-1%29%5E2%2B4%29\"

\n" ); document.write( "The standard method for finding an inverse is to switch the x and y and solve for the new y.

\n" ); document.write( "In terms of the graph, switching the x and y means the inverse is the image of the function reflected in the line y=x.

\n" ); document.write( "Here is a graph of the unrestricted function (red) along with the line y=x (green):

\n" ); document.write( "\"graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C-2%28x-1%29%5E2%2B4%2Cx%29\"

\n" ); document.write( "Here is the formal algebra for finding the inverse function by switching the x and y.

\n" ); document.write( "\"x+=+-2%28y-1%29%5E2%2B4\"
\n" ); document.write( "\"x-4+=+-2%28y-1%29%5E2\"
\n" ); document.write( "\"%28-1%2F2%29%28x-4%29+=+%28y-1%29%5E2\"
\n" ); document.write( "\"sqrt%28%28-1%2F2%29%28x-4%29%29+=+y-1\"
\n" ); document.write( "\"y+=+sqrt%28%28-1%2F2%29%28x-4%29%29%2B1\"

\n" ); document.write( "Here is a graph of the unrestricted function (red) and its inverse (blue), along with the line y=x (green):

\n" ); document.write( "

\n" ); document.write( "You can see that the graph of the inverse function (blue) is the reflection in the line y=x (green) of the portion of the graph of the given function (red) to the right of x=1.

\n" ); document.write( "Now let's go back to the formal algebraic process shown above for finding the inverse by switching the x and y.

\n" ); document.write( "Note that that method of finding an inverse corresponds to viewing the graph of an inverse as the reflection in the line y=x of the given function.

\n" ); document.write( "The inverse can also be found informally, based on the concept that the inverse function \"un-does\" what the function does. To undo what the function does, the inverse function, compared to the given function, has to perform the opposite operations in the opposite order.

\n" ); document.write( "So look at the operations performed on the variable by the given function:

\n" ); document.write( "(1) subtract 1;
\n" ); document.write( "(2) square it;
\n" ); document.write( "(3) multiply by -2; and
\n" ); document.write( "(4) add 4

\n" ); document.write( "Therefore, the inverse function needs to...

\n" ); document.write( "(1) subtract 4: \"x-4\"
\n" ); document.write( "(2) divide by -2 (or multiply by -1/2): \"%28-1%2F2%29%28x-4%29\"
\n" ); document.write( "(3) take the (positive) square root: \"sqrt%28%28-1%2F2%29%28x-4%29%29\"
\n" ); document.write( "(4) add 1: \"sqrt%28%28-1%2F2%29%28x-4%29%29%2B1\"

\n" ); document.write( "That is of course the same result we got using the formal algebra.

\n" ); document.write( "Furthermore, note that the steps for finding the inverse function by this informal method are EXACTLY the same as the steps used in the formal method shown above.

\n" ); document.write( "It is very often the case that the inverse of a relatively simple function can be found in this way much faster than with the formal algebra.

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