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.

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.

Prepp Team · Accepted Answer

Function Analysis for $f(x) = -\frac{|x|}{x}$

The function is defined for non-zero real numbers ($x \neq 0$). We need to analyze its behavior for positive and negative values of $x$.

Case 1: $x > 0$ (Positive x)
When $x$ is positive, the absolute value $|x|$ is equal to $x$.
Therefore, the function becomes: $f(x) = -\frac{x}{x} = -1$. This means for all positive $x$, the function value is constant at $-1$.
Case 2: $x < 0$ (Negative x)
When $x$ is negative, the absolute value $|x|$ is equal to $-x$.
Therefore, the function becomes: $f(x) = -\frac{-x}{x} = -(-1) = 1$. This means for all negative $x$, the function value is constant at $1$.

The function $f(x)$ can be summarized as a piecewise function:

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

This describes a graph consisting of two horizontal line segments:

Since $x$ cannot be zero, there should be open circles at the points $(0, 1)$ and $(0, -1)$ on the graph.

We need to find the plot that matches this description. Let's examine the options:

Option 1 shows a line $y=1$ for $x < 0$ and $y=-1$ for $x > 0$. This matches our analysis.
Option 2 shows the opposite: $y=-1$ for $x < 0$ and $y=1$ for $x > 0$. Incorrect.
Options 3 and 4 show lines passing through the origin with positive and negative slopes, respectively. Incorrect.

Therefore, the plot representing the function $f(x) = -\frac{|x|}{x}$ is the one shown in Option 1.