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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 77
Chapter 3, Problem 77

Graph each function. f(x)={∣x∣if x<36−xif x≥3f(x) =\(\begin{cases}\)|x| & \(\text{if }\) x < 3 \\6 - x & \(\text{if }\) x \(\geq\) 3\(\end{cases}\)

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1
Identify the piecewise function and its two parts: \(f(x) = |4x|\) for \(x < 4\) and \(f(x) = -x + 20\) for \(x \geq 4\).
For the first part, \(f(x) = |4x|\), recognize that the absolute value function outputs the positive value of \$4x\(. So for \)x < 4\(, calculate \)f(x)\( by taking the absolute value of \)4x$.
For the second part, \(f(x) = -x + 20\), this is a linear function with slope \(-1\) and y-intercept \(20\). For \(x \geq 4\), plot this line starting at \(x=4\).
Find the value of the function at the boundary point \(x=4\) for both parts to check continuity: calculate \(|4(4)|\) and \(-4 + 20\) to see if the function values match or if there is a jump.
Plot the graph by drawing the curve for \(f(x) = |4x|\) on the left side of \(x=4\) and the line \(f(x) = -x + 20\) on the right side of \(x=4\), using the boundary values to connect or separate the pieces.

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Key Concepts

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

Piecewise Functions

A piecewise function is defined by different expressions depending on the input value's interval. Understanding how to interpret and graph each piece separately is essential, as the function's rule changes at specified points, called breakpoints.
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Function Composition

Absolute Value Function

The absolute value function, |x|, outputs the distance of x from zero, always non-negative. When graphing, it creates a V-shaped graph, and understanding its behavior helps in plotting parts of the piecewise function involving absolute values.
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Function Composition

Graphing Linear Functions

Linear functions have the form f(x) = mx + b and graph as straight lines. For the piecewise function, each piece is linear or absolute value-based, so knowing how to plot lines and interpret slopes and intercepts is crucial for accurate graphing.
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Graphs of Logarithmic Functions
Related Practice
Textbook Question

If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. (-1, -3), (-5, 12), (1, -11)

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Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.

ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=x^3, g(x)=x^2+3x-1

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Textbook Question

Given functions f and g, (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=x^3, g(x)=x^2+3x-1

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Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.

ƒ(x)=x+2,g(x)=x4+x2−4ƒ(x)=x+2, g(x)=x^4+x^2-4

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Textbook Question

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

y=∣x−1∣y = | x - 1 |

y=x2−4y = x^2 - 4

How is the graph of y = x^2 - 4 obtained by transforming the graph of y=x2y = x^2?

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Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√x, g(x)=x-1

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