Function Analysis and Value Calculation

The given function is defined as $f(y)=|\ y\ |/\ y$. This function behaves differently based on the sign of $y$. Since $q$ is a non-zero real number, we consider two cases for $q$.

Understanding the Function $f(y)$

• If $y > 0$, then $|y| = y$. So, $f(y) = y/y = 1$.
• If $y < 0$, then $|y| = -y$. So, $f(y) = (-y)/y = -1$.

This means $f(y)$ returns $1$ for positive inputs and $-1$ for negative inputs.

Calculating $|f(q) - f(-q)|$

We need to find the value of $|\ f(q)\ –\ f(-q)\ |$. Let's analyze based on the sign of $q$.

Case 1: $q > 0$

• Since $q > 0$, $f(q) = 1$.
• Also, $-q < 0$, so $f(-q) = -1$.
• Therefore, $|\ f(q)\ –\ f(-q)\ | = |\ 1\ –\ (-1)\ | = |\ 1 + 1\ | = |\ 2\ | = 2$.

Case 2: $q < 0$

• Since $q < 0$, $f(q) = -1$.
• Also, $-q > 0$, so $f(-q) = 1$.
• Therefore, $|\ f(q)\ –\ f(-q)\ | = |\ (-1)\ –\ 1\ | = |\ -2\ | = 2$.

Conclusion

In both possible cases for a non-zero real number $q$, the value of $|\ f(q)\ –\ f(-q)\ |$ is $2$.