Evaluate if possible the function ar the specific value of independent variable and simplify the result f(x)=|x|/x a(f(2)) b(f(-2(( c(f(x^2))d(f(x-))

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.

Evaluate if possible the function at the specific value of the independent variable and simplify the result f(x)=|x|/x a(f(2)) b(f(-2)) c(f(x^2)) d(f(x-))

Learn how to evaluate the function f(x) = |x|/x at specific values of x, including f(2), f(-2), f(x^2), and analyze limits such as f(x-). This problem covers key mathematical concepts, including absolute value, piecewise functions, and limits. Suitable for students in Grades 10-12.

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