Let's break down and evaluate the given function $f(x) = \frac{|x|}{x}$ at the specified values of the independent variable.

Step 1: Understanding the function $f(x) = \frac{|x|}{x}$

• This function compares the absolute value of $x$ (which is always non-negative) with the original value of $x$.
• Specifically:
  • If $x > 0$, then $f(x) = 1$ (since $\frac{|x|}{x} = \frac{x}{x} = 1$).
  • If $x < 0$, then $f(x) = -1$ (since $\frac{|x|}{x} = \frac{-x}{x} = -1$).
  • If $x = 0$, then the function is undefined (division by zero is not allowed).

Now, let's evaluate this function at each of the given points:

Part (a): $f(f(2))$

1. First, evaluate $f(2)$:
   • Since $2 > 0$, we have $f(2) = 1$.
2. Now, evaluate $f(1)$:
   • Since $1 > 0$, we have $f(1) = 1$.

So, $f(f(2)) = 1$.

Part (b): $f(-2)$

Evaluate $f(-2)$:

• Since $-2 < 0$, we have $f(-2) = -1$.

So, $f(-2) = -1$.

Part (c): $f(x^2)$

Evaluate $f(x^2)$:

• Since $x^2 \geq 0$ for all $x$ (because squaring any real number makes it non-negative), we have $f(x^2) = 1$ for all values of $x$ (except $x = 0$, where the function is undefined).

So, $f(x^2) = 1$ for all $x \neq 0$.

Part (d): $f(x-)$

This seems like an incomplete expression or could refer to the left-hand limit $\lim_{x \to a^-} f(x)$, but without more information, it's unclear what this represents. Could you clarify this part?

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Do you want further details or have any questions?

Here are five related questions:

1. How does the absolute value function affect the behavior of $f(x) = \frac{|x|}{x}$?
2. What happens to $f(x) = \frac{|x|}{x}$ as $x$ approaches 0?
3. Can the function $f(x)$ be modified to handle the case when $x = 0$?
4. How would $f(x)$ behave if $x$ were a complex number?
5. What is the piecewise definition of $f(x)$?

Tip: When dealing with absolute value functions, always check the sign of the variable inside to simplify the function properly.