Given the function $f(x) = \frac{|x|}{x}$, let's evaluate it for the given scenarios:

(a) $f(2)$

Here, $x = 2$. Since 2 is positive: $f(2) = \frac{|2|}{2} = \frac{2}{2} = 1$

(b) $f(-2)$

Here, $x = -2$. Since -2 is negative: $f(-2) = \frac{|-2|}{-2} = \frac{2}{-2} = -1$

(c) $f(x^2)$

Here, $x^2$ is always non-negative (regardless of the value of $x$): $f(x^2) = \frac{|x^2|}{x^2} = \frac{x^2}{x^2} = 1$

(d) $f(x-1)$

Here, the function depends on whether $x-1$ is positive or negative:

• If $x > 1$, then $x-1$ is positive, so $f(x-1) = 1$.
• If $x < 1$, then $x-1$ is negative, so $f(x-1) = -1$.
• If $x = 1$, then $x-1 = 0$, and the function is undefined because division by zero occurs.

Would you like more details or have any questions?

Here are some related questions to explore:

1. How does $f(x)$ behave for different values of $x$?
2. What is the graph of $f(x) = \frac{|x|}{x}$ and how does it look?
3. How would $f(x) = \frac{|x|}{x}$ be defined if $x = 0$ were included?
4. What happens when you compose $f(x)$ with another function, like $g(x) = x^2$?
5. Can $f(x) = \frac{|x|}{x}$ be expressed as a piecewise function?

Tip: Functions like $f(x) = \frac{|x|}{x}$ are useful in defining the sign of a real number, commonly known as the "signum" function.