f(x)=|x|/x (a)f(2) (b)f(-2) (c)f(x2) (d)f(x-1)

The function \( f(x) = \frac{|x|}{x} \) is known as the sign function or the signum function, and it outputs the sign of \( x \). The function can be defined piecewise as follows:

\[
f(x) =
\begin{cases} 
1 & \text{if } x > 0 \\
-1 & \text{if } x < 0 \\
\text{undefined} & \text{if } x = 0 
\end{cases}
\]

Let's solve each part:

### (a) \( f(2) \)
Since \( 2 > 0 \), we have:
\[
f(2) = 1
\]

### (b) \( f(-2) \)
Since \( -2 < 0 \), we have:
\[
f(-2) = -1
\]

### (c) \( f(x^2) \)
Since \( x^2 \) is always non-negative, \( x^2 \geq 0 \), and it is positive when \( x \neq 0 \). Therefore:
\[
f(x^2) =
\begin{cases} 
1 & \text{if } x \neq 0 \\
\text{undefined} & \text{if } x = 0
\end{cases}
\]

### (d) \( f(x-1) \)
This function depends on whether \( x-1 \) is positive, negative, or zero:

\[
f(x-1) =
\begin{cases} 
1 & \text{if } x > 1 \\
-1 & \text{if } x < 1 \\
\text{undefined} & \text{if } x = 1
\end{cases}
\]

Would you like further details on any of these parts or have any additional questions?

Here are 5 related questions you might find interesting:
1. How does the sign function behave as \( x \) approaches zero?
2. Can \( f(x) = \frac{|x|}{x} \) be differentiated, and if so, where is it differentiable?
3. What is the significance of the function \( f(x) = \frac{|x|}{x} \) in physics?
4. How does the function \( f(x) = \frac{|x|}{x} \) compare to the Heaviside step function?
5. How would \( f(x) \) behave if we added a constant term to \( |x| \), like \( f(x) = \frac{|x| + c}{x} \)?

**Tip:** When analyzing piecewise functions, it's important to consider the definition in each interval and check the behavior at the boundaries.

Explore the sign function f(x) = |x|/x and calculate its values at specific points such as f(2), f(-2), f(x^2), and f(x-1). Understand the piecewise definition of the function and its behavior in different scenarios.

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