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.

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

All textbooks

Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 77

Chapter 6, Problem 77

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

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} $.

Identify the expression inside the absolute value: here, $ A = x - 1 $.

Set up the piecewise function based on the sign of $ x - 1 $:

\[ y = |x - 1| = \begin{cases} x - 1 & \text{if } x - 1 \geq 0 \\ -(x - 1) & \text{if } x - 1 < 0 \end{cases} \]

Simplify the inequalities and expressions inside the piecewise function:

\[ y = \begin{cases} x - 1 & \text{if } x \geq 1 \\ -x + 1 & \text{if } x < 1 \end{cases} \]

This piecewise function now represents $ y = |x - 1| $ without the absolute value symbol.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

Guided course

3:21

Nonlinear Inequalities

Related Practice

Textbook Question

Perform each operation, if possible.

$[\begin{matrix} 2 & 5 & 8 \\ 1 & 9 & 2 \end{matrix}] - [\begin{matrix} 3 & 4 \\ 7 & 1 \end{matrix}]$

Textbook Question

The graphs show regions of feasible solutions. Find the maximum and minimum values of each objective function. objective function = 3x + 5y

views

Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order.

y = | x - 1 |

y = x² - 4

How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

Textbook Question

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions?

x - 2y = 3

-2x + 4y = k

Textbook Question

Use a system of equations to solve each problem. See Example 8. Find an equation of the line y = ax + b that passes through the points (-2, 1) and (-1, -2).

Textbook Question

Perform each operation, if possible.

$[\begin{matrix} - 1 & 0 \\ 2 & 5 \end{matrix}] [\begin{matrix} - 3 & 4 \\ 2 & 8 \end{matrix}]$