Consider the following nonlinear system. Work Exercises 75 –80 in order.
y = | x - 1 |
y = x² - 4
Use the definition of absolute value to write y = | x - 1 | as a piecewise-defined function.

Verified step by step guidance

1. Recall the definition of the absolute value function: for any expression \( A \), \( |A| = \begin{cases} A & \text{if } A \geq 0 \\ -A & \text{if } A < 0 \end{cases} \).
2. Identify the expression inside the absolute value: here, \( A = x - 1 \).
3. Set up the piecewise function based on the sign of \( x - 1 \):
4. \[ y = |x - 1| = \begin{cases} x - 1 & \text{if } x - 1 \geq 0 \\ -(x - 1) & \text{if } x - 1 < 0 \end{cases} \]
5. Simplify the inequalities and expressions inside the piecewise function:
6. \[ y = \begin{cases} x - 1 & \text{if } x \geq 1 \\ -x + 1 & \text{if } x < 1 \end{cases} \]
7. This piecewise function now represents \( y = |x - 1| \) without the absolute value symbol.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always non-negative. For an expression like |x - 1|, it equals x - 1 when x - 1 is non-negative, and -(x - 1) when x - 1 is negative. This definition allows rewriting absolute value expressions as piecewise functions.

Piecewise-Defined Functions

A piecewise-defined function is expressed using different formulas over different intervals of the domain. It is useful for representing functions like absolute value, where the rule changes based on the input value. Understanding how to write and interpret these functions is essential for analyzing nonlinear systems.

Nonlinear Systems of Equations

A nonlinear system involves equations where variables are raised to powers other than one or involve absolute values. Solving such systems often requires rewriting expressions (like absolute values) and analyzing intersections of curves, such as y = |x - 1| and y = x^2 - 4, to find common solutions.