Graphing the Function | x y x 1

The equation x y x 1 may seem complex at first glance, but by breaking it down into manageable parts, we can efficiently determine and graph its behavior. Let's explore this step-by-step.

Understanding the Equation

The given equation is x y x 1. To graph this, we need to consider how the absolute value of x and y affects the solution. This can be done by analyzing various cases based on the value of x.

Step-by-Step Analysis

Case 1: x ≥ 0

When x is non-negative:

Since |x| x, the equation simplifies to:

x y x 1

Solving for y, we get:

y 1

For this case, y is a constant value of 1, regardless of the value of x. This results in a horizontal line at y 1.

Additionally, we also consider the case where:

y -1. This again results in a horizontal line at y -1.

Case 2: x 0

When x is negative:

Since |x| -x, the equation becomes:

-x y x 1

Reorganizing the equation, we get:

y 2x 1

However, since x 0, we need to consider sub-cases:

Sub-case 2a: 2x 1 ≥ 0

In this sub-case, x ≥ -1/2. This implies:

y 2x 1

This is a straight line with a slope of 2, intersecting the y-axis at (0, -1).

Sub-case 2b: 2x 1 0

In this sub-case, x -1/2. This implies:

y -2x 1

This is another straight line with a slope of -2, intersecting the y-axis at (0, -1).

Combining Results and Graphing

By combining the results from both cases, we have:

For x ≥ 0, we have two horizontal lines: y 1 and y -1. For x 0 and -1/2 ≤ x 0, we have two lines: y 2x 1 and y -2x 1. These lines intersect the y-axis at (0, -1) and the x-axis at (-1/2, 0).

The final graph will consist of:

Two horizontal lines: y 1 and y -1. Two lines: y 2x 1 and y -2x 1.

To visualize, here’s a rough sketch of the graph:

|y|

1  ------------------- for x  0
0 ---------------------------------------- x
-1 -----------------------   for x  0

The horizontal lines are at y 1 and y -1 for x ≥ 0. The lines for x 0 slope up and down depending on the values derived.

This completes the graphing of the equation x y x 1.