Determine whether each of these functions from Z to Z is one-to-one.
a) f(x) = x−2
b) f(x) = x2 −1
c) f(x) = x3
a:f(x)=f(y)⇒x−2=y−2⇒x=y − yesb:f(1)=12−1=0f(−1)=(−1)2−1=0f(1)=f(−1) − noc:f(x)=f(y)⇒x3=y3⇒x=y − yesa:\\f\left( x \right) =f\left( y \right) \Rightarrow x-2=y-2\Rightarrow x=y\,\,-\,\,yes\\b:\\f\left( 1 \right) =1^2-1=0\\f\left( -1 \right) =\left( -1 \right) ^2-1=0\\f\left( 1 \right) =f\left( -1 \right) \,\,-\,\,no\\c:\\f\left( x \right) =f\left( y \right) \Rightarrow x^3=y^3\Rightarrow x=y\,\,-\,\,yesa:f(x)=f(y)⇒x−2=y−2⇒x=y−yesb:f(1)=12−1=0f(−1)=(−1)2−1=0f(1)=f(−1)−noc:f(x)=f(y)⇒x3=y3⇒x=y−yes
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