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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 24
Chapter 3, Problem 24

Graph each piecewise-defined function.
f(x)={2x+1if x≥0xif x<0f(x) =\(\begin{cases}\)2x + 1 & \(\text{if }\) x \(\geq\) 0 \(\x\) & \(\text{if }\) x < 0\(\end{cases}\)

Verified step by step guidance
1
Identify the two pieces of the piecewise function: \(f(x) = 4x - 2\) for \(x \leq 0\) and \(f(x) = 2x\) for \(x > 0\).
For the first piece \(f(x) = 4x - 2\) when \(x \leq 0\), find key points by substituting values of \(x\) less than or equal to 0, such as \(x = 0\) and \(x = -1\).
Plot the points from the first piece on the coordinate plane and draw a line through them, making sure to include the point at \(x=0\) with a solid dot since the inequality includes equality (\(\leq\)).
For the second piece \(f(x) = 2x\) when \(x > 0\), find points by substituting values of \(x\) greater than 0, such as \(x = 1\) and \(x = 2\).
Plot these points and draw a line through them, using an open circle at \(x=0\) to indicate that this point is not included in the second piece since the inequality is strict (\(>\)).

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Key Concepts

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

Piecewise-Defined Functions

A piecewise-defined function is a function composed of different expressions depending on the input value's domain. Each piece applies to a specific interval, and the function's overall graph is formed by combining these pieces. Understanding how to interpret and graph each piece separately is essential.
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Domain Restrictions of Composed Functions

Graphing Linear Functions

Each piece of the function is linear, represented by an equation of the form y = mx + b. To graph these, identify the slope (m) and y-intercept (b), then plot points accordingly. For piecewise functions, graph each linear piece only over its specified domain.
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Graphs of Logarithmic Functions

Domain Restrictions and Continuity

The domain restrictions (e.g., x ≤ 0 or x > 0) determine where each piece of the function applies. When graphing, ensure to respect these boundaries, using closed or open dots to indicate inclusion or exclusion of endpoints. This helps in understanding the function's continuity and behavior at boundary points.
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Domain Restrictions of Composed Functions
Related Practice
Textbook Question

For the pair of functions defined, find (ƒ+g)(x), (ƒ-g)(x), (ƒg)(x), and (f/g)(x).Give the domain of each. ƒ(x)=√(5x-4), g(x)=-(1/x)

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Textbook Question

For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2.

ƒ(x)=√(4x-1), g(x)=1/x

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Textbook Question

Determine whether each relation defines a function, and give the domain and range.

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Textbook Question

Determine whether the three points are the vertices of a right triangle. See Example 3.

(−2,−8),(0,−4),(−4,−7)(-2,-8),(0,-4),(-4,-7)

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Textbook Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Textbook Question

For the pair of functions defined, find (f/g)(x).Give the domain of each. See Example 2.

ƒ(x)=√(4x-1), g(x)=1/x

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