Question

Which one of the following plots represents \( f(x) = -\frac{|x|}{x} \), where \( x \) is a non-zero real number?
Note: The figures shown are representative.

• A.
• B.
• C.
• D.

Hint:

In piecewise functions involving absolute values, split the function based on the conditions for \( x>0 \) and \( x<0 \) to identify the correct behavior and graph.

Answer 1

The Correct Option is A

Step 1: Analyze the function.
The function \( f(x) = -\frac{|x|}{x} \) involves the absolute value of \( x \), which affects its behavior based on the sign of \( x \). The function can be rewritten as:
\[ f(x) = \begin{cases} -1 & \text{if } x > 0 \\ \ \ 1 & \text{if } x < 0 \end{cases} \] Thus, for \( x > 0 \), \( f(x) = -1 \), and for \( x < 0 \), \( f(x) = 1 \).

Step 2: Identify the correct graph.
From the given function, we see that the graph will be a piecewise constant function:
For \( x > 0 \), the function value is \( -1 \), so the graph will be a horizontal line at \( f(x) = -1 \) for positive \( x \).
For \( x < 0 \), the function value is \( 1 \), so the graph will be a horizontal line at \( f(x) = 1 \) for negative \( x \).

Step 3: Compare with the options.
Option (A) matches this behavior, where for \( x > 0 \), \( f(x) = -1 \), and for \( x < 0 \), \( f(x) = 1 \). The graph shows this exact pattern, making it the correct choice.