Question 125353
If x is negative, |x|=-x

If x is zero or positive, |x|=x


{{{f(x)=abs(x)/x}}}

So,

{{{f(-2)=abs(2)/-2=2/-2=-1}}}
{{{f(-1)=abs(-1)/-1=1/-2=-1}}}
{{{f(0)=abs(0)/0=0/0}}}-----this is undefined.
Evidence?
{{{graph(1000,1000,-20,20,-20,20,abs(x)/x)}}}
There is a discontinuity at x=0.
{{{f(5)=abs(5)/5=5/5=1}}}
Please take note that if x is a real number then x^2 is <b>always</b> positive.
So,
{{{f(x^2)=abs(x^2)/x^2=x^2/x^2=1}}}
IF x is positive, 1/x is positive
If x is zero, 1/x is undefined
If x is negative, 1/x is negative

So,
If x is negative,
{{{f(1/x)=abs(1/x)/(1/x)=(-1/x)/(1/x)=-1}}}
If x is zero,
{{{f(1/0)}}}-----you can see the result, it is undefined
If x is positive,
{{{f(1/x)=abs(1/x)/(1/x)=(1/x)/(1/x)=1}}}


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