Question

Let f(x) = |x| / x , where x can be any real number except 0.

Answer 1

Answer: Hello there!

we have the function f(x) = IxI/x

Then this function can be piecewise writen as:

f(x) = 1 if x > 0

f(x) = -1 if x < 0

because:

suposse that n is a number greater than 0, then:

f(n) = InI/n = n/n = 1

and this is independent of the number n, so for all the positive values of x, we have f(x) = 1

and

f(-n) = I-nI/(-n) = n/(-n) = -1

and again, this does not depend on the value of n, so for all the negative values of x, we have f(x) = -1.

On the other hand, the Heaviside function is defined as:

H(x) = 1 if x>0

H(x) = 0 if x<0

Then the difference is that, when the Heaviside function takes the value of 0 for negative values of x, our function f(x) takes the value of -1 for negative values of x.