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You can put this solution on YOUR website!
\n" ); document.write( "You don't give us any idea of what you know about inverses of functions, so I'm not sure what to write in my response....
\n" ); document.write( "part 1: functions that are inverses of each other.
\n" ); document.write( "Every linear polynomial is a function that has an inverse. For example, consider the function y = 3x-2.
\n" ); document.write( "An algebraic way to find the inverse of a function is to switch the x and y variables and solve for the new y. For this linear function, it looks like this:
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\n" ); document.write( "Another way to find the inverse of a given linear function is to use the fact that the inverse function \"undoes\" what the function does -- that is, an inverse function \"gets you back where you started\".
\n" ); document.write( "To find the inverse function for this example by this method, observe that the function does the following to the input value:
\n" ); document.write( "(1) multiply by 3; and
\n" ); document.write( "(2) subtract 2
\n" ); document.write( "The inverse function, to get you back where you started, has to do the opposite operations in the reverse order; for this example,
\n" ); document.write( "(1) add 2; and
\n" ); document.write( "(2) divide by 3
\n" ); document.write( "Applying those operations to the input value x gives you the inverse function:
\n" ); document.write( "So now you have a way of finding an infinite number of pairs of functions that are inverses of each other.
\n" ); document.write( "Do you know that the graphs of two function that are inverses of each other are mirror images of each other with respect to the line y=x? That should make sense, since the algebraic method for finding the inverse of a function is to switch the x and y variables. In terms of the graphs of the functions, switching x and y is equivalent to reflecting the graph about the line y=x. (In reflecting a graph about the line y=x, the x axis becomes the y axis, and vice versa; so the x and y are switched.)
\n" ); document.write( "Here are graphs of the function (red) in the preceding example and its inverse (green), and of the line y=x (blue). You can see that the function and its inverse are mirror images of each other.
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\n" ); document.write( "part 2: functions that are their own inverses.
\n" ); document.write( "You can use the fact that the graphs of a function and its inverse are mirror images of each other in the line y=x to find an infinite number of functions that are their own inverses. If a function is its own inverse, then the reflection of its graph in the line y=x must be the same line; the graph must have symmetry with respect to the line y=x.
\n" ); document.write( "Here are the graphs of the functions you note in your message that are their own inverses: y = -x (red), y=1/x (green), and y=-1/x (blue), along with, again the graph of y=x (purple). Note that, although you can't see it like with the other three, the graph shows that y=x is also its own inverse.
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\n" ); document.write( "Now you get to finish this second part of your problem yourself, using the idea of a function that is its own inverse being symmetrical about the line y=x:
\n" ); document.write( "Do you see from the graph what other linear functions, similar to y=-x, will have symmetry with respect to the line y=x?
\n" ); document.write( "And what about other functions similar to y=1/x and y=-1/x that will have symmetry with respect to the line y=x? \n" ); document.write( "