Question

Find the range of function f(x)=x/|x|​

Answer 1

Answer:

I'll be breaking down this question in three parts,

When x is positive

When x is negative

When x is zero

In the first case, if x is positive,

And since the modulus of a positive number is the number itself, our function becomes,

f(x) = x/x = 1, here I have replaced modulus of x, by x, because there is no change.

In the second case, when x is negative, the value of our function changes to,

f(x) = -x/x = - 1. Because the numerator stays the same, however our denominator has the modulus function which turns it into its positive counterpart. So while the numerator is (-x), our denominator has already become (+x).

And the third case is invalid, because 0/0 is not defined.

Hence our domain becomes (-infinte, +infinite) - {0}

And our range would be, the only two possible solutions, {-1, 1}.

Answer 2

if x is positive

And since the modulus of a positive number is the number itself, our function becomes,

f(x) = x/x = 1, replaced |x|, by x,

In the second case, when x is negative, the value of our function changes to,

f(x) = x/-x = - 1.

If x is zero

as 0/0 is not defined.

Hence our domain becomes (-infinte, +infinite) - {0}

And our range would be, the only two possible solutions, {-1, 1}.