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Ch. R - Algebra Review
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. R - Algebra ReviewProblem 65
Chapter 1, Problem 65

Graph each function. See Examples 6–8. g(x) = |x| - 1

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Recognize that the function is given by \(g(x) = |x| - 1\), which involves the absolute value function \(|x|\). The absolute value function outputs the distance of \(x\) from zero, so it is always non-negative.
Recall the basic shape of the graph of \(y = |x|\): it forms a 'V' shape with its vertex at the origin \((0,0)\), where the graph is symmetric about the y-axis.
To graph \(g(x) = |x| - 1\), start by taking the graph of \(y = |x|\) and then shift it downward by 1 unit because of the '-1' outside the absolute value.
Plot key points to help sketch the graph: for example, at \(x=0\), \(g(0) = |0| - 1 = -1\); at \(x=1\), \(g(1) = |1| - 1 = 0\); and at \(x=-1\), \(g(-1) = |-1| - 1 = 0\). These points help define the vertex and the arms of the 'V'.
Draw the graph by connecting these points with two straight lines forming a 'V' shape, with the vertex at \((0, -1)\) and the arms extending upward with slope 1 to the right and slope -1 to the left.

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Key Concepts

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

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x, making all inputs positive or zero. Its graph forms a 'V' shape with the vertex at the origin, reflecting negative inputs across the y-axis.
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Function Transformation

Function transformations involve shifting, stretching, or reflecting graphs. For g(x) = |x| - 1, subtracting 1 shifts the entire graph of |x| downward by one unit, moving the vertex from (0,0) to (0,-1).
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Domain and Range of Function Transformations

Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise functions, defining different expressions for x ≥ 0 and x < 0. Understanding this helps in plotting points accurately and visualizing the graph's shape.
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Related Practice
Textbook Question

Factor each polynomial completely. See Example 6. 40ab - 16a

Textbook Question

Use the graph of y = ƒ(x) to find each function value: (a) ƒ(-2) (b) ƒ(0) (c) ƒ(1) and (d) ƒ(4). See Example 7(d).

Textbook Question

Add or subtract, as indicated. See Example 4. 4/(x+1) + 1/(x² - x + 1) - 12/(x³ + 1)

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Textbook Question

For each function, find (a) ƒ(2) and (b) ƒ(-1). See Example 7.

Textbook Question

Graph each function. See Examples 6–8. h(x) = -(x + 1)³

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Textbook Question

Use the product and quotient rules for radicals to rewrite each expression. See Example 4. √5 /√20

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