Consider the following nonlinear system. Work Exercises 75 –80 in order.y = | x - 1 |y = x^2 - 4Use the definition of absolute value to write y = | x - 1 | as a piecewise-defined function.

Verified step by step guidance

1. Understand that the absolute value function \( y = |x - 1| \) can be expressed as a piecewise function because it behaves differently based on the value of \( x \).
2. Recall the definition of absolute value: \( |a| = a \) if \( a \geq 0 \) and \( |a| = -a \) if \( a < 0 \).
3. Apply this definition to \( y = |x - 1| \):
4. For \( x - 1 \geq 0 \), which simplifies to \( x \geq 1 \), the function is \( y = x - 1 \).
5. For \( x - 1 < 0 \), which simplifies to \( x < 1 \), the function is \( y = -(x - 1) = -x + 1 \).

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of a number x from zero on the number line, always yielding a non-negative result. For any real number x, |x| is defined as x if x is greater than or equal to zero, and -x if x is less than zero. This property is crucial for transforming the absolute value equation into a piecewise function.

Piecewise Functions

A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the function's domain. In the context of the absolute value function, it allows us to express the function differently based on the value of x. For example, the function y = |x - 1| can be expressed as y = x - 1 for x ≥ 1 and y = -(x - 1) for x < 1.

Nonlinear Systems

A nonlinear system consists of equations that do not form a straight line when graphed, often involving polynomial, exponential, or absolute value functions. In this case, the system includes the absolute value function and a quadratic function, y = x^2 - 4. Understanding how to analyze and solve such systems is essential for finding points of intersection and understanding their graphical behavior.