document.write( "Question 805752: Determine which function(s) below have an inverse f^(-1). Can you please explain to me why?\r
\n" ); document.write( "\n" ); document.write( "(I) f(x)=x^2-2x (II) f(x)=1/x (III) f(x)=cos x (IV) f(x)=sin x,-π/2≤x≤π/2\r
\n" ); document.write( "\n" ); document.write( "(A) I & II\r
\n" ); document.write( "\n" ); document.write( "(B) I, II, & III\r
\n" ); document.write( "\n" ); document.write( "(C) II, III, & IV\r
\n" ); document.write( "\n" ); document.write( "(D) II & IV\r
\n" ); document.write( "\n" ); document.write( "(E) none of the above \n" ); document.write( "

Algebra.Com's Answer #485453 by KMST(5328) $\"\"$ $\"About$

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A relation between x and y establishes y as a function of x if for each x we find no more that one y paired to that x.
\n" ); document.write( "It could be that y is a function of x that happens to assign the same value of y to more than one x.
\n" ); document.write( " $\"y=x%5E2\"$ is an example. $\"graph%28200%2C200%2C-3%2C3%2C-1%2C9%2Cx%5E2%29\"$
\n" ); document.write( "Each x has one and only one $\"x%5E2\"$ ,
\n" ); document.write( "but the same y value can be the y for more than one x,
\n" ); document.write( "for example $\"y=4\"$ is the y for $\"x=2\"$ and for $\"x=-2\"$ .
\n" ); document.write( "In that case, we cannot reverse the function and sat that x is established as a function of y, because for y=4 we find two corresponding values of x.
\n" ); document.write( "If there is just one x (or less) for each y, then we can solve for x and find the reverse function.
\n" ); document.write( "
\n" ); document.write( "(I) $\"f%28x%29=x%5E2-2x\"$ $\"graph%28200%2C200%2C-2%2C4%2C-2%2C8%2Cx%5E2-2x%29\"$
\n" ); document.write( " $\"f%280%29=0=f%282%29\"$ so $\"x=0\"$ and $\"x=2\"$ have the same $\"y=0\"$ ----> There is no inverse function
\n" ); document.write( "(II) $\"f%28x%29=1%2Fx\"$ can be written as $\"y=1%2Fx\"$ $\"graph%28200%2C200%2C-2%2C8%2C-2%2C8%2C1%2Fx%29\"$
\n" ); document.write( "Exchanging the places of the variables we get $\"x=1%2Fy\"$ --> $\"y=1%2Fx\"$
\n" ); document.write( "So $\"y=1%2Fx\"$ or $\"g%28x%29=1%2Fx%29\"$ is the inverse function of $\"f%28x%29=1%2Fx\"$
\n" ); document.write( "That function has an inverse, and is its own inverse.
\n" ); document.write( "(III) $\"f%28x%29=cos+x\"$ $\"graph%28300%2C100%2C-1.5%2C7.5%2C-1.5%2C1.5%2Ccos%28x%29%29\"$
\n" ); document.write( "We know that cosine is a periodic function with period $\"2pi\"$ so the of $\"y=f%28x%29=cos%28x%29\"$ repeat at $\"2pi\"$ intervals so that $\"cos%280%291=cos%282pi%29=cos%284pi%29\"$ $\"%22=+....%22\"$ so we would not know what value of x to assign to y=1.
\n" ); document.write( " $\"f%28x%29=cos+x\"$ does not have an inverse function
\n" ); document.write( "(IV) $\"f%28x%29=sin%28x%29\"$ , $\"-pi%2F2%3C=x%3C=pi%2F2\"$ has a restricted range.
\n" ); document.write( " $\"f%28-pi%2F2%29=-1\"$ and as x increases $\"f%28x%29=sin+x\"$ increases, all the way to $\"f%28pi%2F2%29=sin+%28pi%2F2%29=1\"$ , without repeating any values.
\n" ); document.write( "For $\"pi%2F2%3Cx%3C3pi%2F2\"$ the values of $\"sin%28x%29\"$ decrease from 1 all the way to -1, repeating values already found for $\"sin%28x%29\"$ in the $\"%28matrix%281%2C3%2C+-pi%2F2+%2C%22%2C%22%2C+pi%2F2%29+%29\"$ interval.
\n" ); document.write( "However, with $\"f%28x%29\"$ defined as $\"f%28x%29=sin%28x%29\"$ only in the restricted domain $\"-pi%2F2%3C=x%3C=pi%2F2\"$ ,
\n" ); document.write( "each x corresponds to one and only one y, and vice versa.
\n" ); document.write( "The function has an inverse.
\n" ); document.write( "
\n" ); document.write( "So the answer is $\"highlight%28%28D%29%29\"$ II and IV only \n" ); document.write( "

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