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

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.

Explore how to evaluate the function f(x) = |x|/x with detailed solutions for various inputs, including f(2), f(-2), f(x^2), and f(x-1). This guide covers essential algebraic concepts like the absolute value, signum function, and piecewise functions. Ideal for high school students in Grades 9-12.

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