document.write( "Question 629914: two function f and g are defined by :\r
\n" ); document.write( "\n" ); document.write( "f(x)=2x+1
\n" ); document.write( "g(x)=x^2\r
\n" ); document.write( "\n" ); document.write( "verify that (fog)^-1 = g^-1 o f^-1 \n" ); document.write( "
Algebra.Com's Answer #396654 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Inverses of of functions are found by swapping the x's and y's. To perform this swap, I like to temporarily replace the function notation with a \"y\":
\n" ); document.write( "f(x) = 2x+1
\n" ); document.write( "y = 2x + 1
\n" ); document.write( "Swap the x's and y's:
\n" ); document.write( "x = 2y + 1 <=== This is the inverse, but not in a desired form
\n" ); document.write( "Solve for y:
\n" ); document.write( "x - 1 = 2y
\n" ); document.write( "Divide by 2 (or multiply by 1/2):
\n" ); document.write( "\"%281%2F2%29x+-+1%2F2+=+y\"
\n" ); document.write( "\"%281%2F2%29x+-+1%2F2+=+f%5E%28-1%29%28x%29\"

\n" ); document.write( "Doing the same for g(x):
\n" ); document.write( "\"g%28x%29+=+x%5E2\"
\n" ); document.write( "\"y+=+x%5E2\"
\n" ); document.write( "\"x+=+y%5E2\"
\n" ); document.write( "\"sqrt%28x%29+=+sqrt%28y%5E2%29\"
\n" ); document.write( "+\"sqrt%28x%29+=+y\"
\n" ); document.write( "Note: Because of the +, the inverse of g is not a function (i.e. some, in fact all, x's have more than one y value). And since this inverse is not a function, I do not believe it is proper to use function notation, g^-1 on it.

\n" ); document.write( "For the compositions it helps if we have a good understanding of what the function's definition is telling you. For example
\n" ); document.write( "f(x) = 2x+1
\n" ); document.write( "The left side tells us that the input to the function, between the parentheses after the function name, is being called \"x\". \"x\" is just a placeholder. It is just being used as a name for the input to the function. We could use any letter here to name the input. f(q) = 2q+1 is the exact same function as f(x)!!

\n" ); document.write( "The right side tells us what function f does with its input. It multiplies the input, x, by 2 and then adds 1. Since \"x\" is just a place holder, function f will take any input, multiply it by 2 and then add 1:
\n" ); document.write( "f(7) = 2(7) + 1
\n" ); document.write( "f(1002.4) = 2(1002.4) + 1
\n" ); document.write( "f(12x-3) = 2(12x-3) + 1
\n" ); document.write( "etc.
\n" ); document.write( "(fog)(x) is just another way to write f(g(x)). In f's parentheses we have \"g(x)\". So \"g(x)\" is the input to f. And what does f do to its input? Answer: it multiplies by 2 and then adds 1!:
\n" ); document.write( "f(g(x)) = 2(g(x)) + 1
\n" ); document.write( "Since \"g%28x%29+=+x%5E2\" we can substitute this in for g(x):
\n" ); document.write( "\"f%28g%28x%29%29+=+2%28g%28x%29%29+%2B+1+=+2%28x%5E2%29+%2B+1\"

\n" ); document.write( "Now we find the inverse of fog:
\n" ); document.write( "\"y+=+2x%5E2+%2B+1\"
\n" ); document.write( "\"x+=+2y%5E2+%2B+1\"
\n" ); document.write( "\"x+-+1+=+2y%5E2\"
\n" ); document.write( "\"%281%2F2%29x+-+1%2F2%29+=+y%5E2\"
\n" ); document.write( "+\"sqrt%28%281%2F2%29x+-+1%2F2%29+=+y\"
\n" ); document.write( "Again, we get an inverse that is not a function.

\n" ); document.write( "For g^-1 o f^-1, we use the two inverses we found back at the start and feed the inverse of f into the inverse of g as its input: From the inverse of g, +\"sqrt%28x%29+=+y\", we can see that it finds the positive and negative square roots of its input. So it will do the same to f^1, \"%281%2F2%29x+-+1%2F2\":
\n" ); document.write( "+\"sqrt%28%281%2F2%29x+-+1%2F2%29+=+y\"
\n" ); document.write( "which matches the inverse of fog! \n" ); document.write( "
\n" );