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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 28
Chapter 3, Problem 28

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.
Graph of a quadratic function with vertex at (8, -2) in college algebra functions chapter.

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Step 1: Determine if the relation is a function by using the vertical line test. If any vertical line intersects the graph at more than one point, the relation is not a function. For this graph, each vertical line intersects the parabola at exactly one point, so it defines a function.
Step 2: Identify the domain of the function. The domain consists of all possible x-values for which the function is defined. Since the parabola extends infinitely to the left and right, the domain is all real numbers, which can be written as \((-\infty, \infty)\).
Step 3: Identify the range of the function. The range consists of all possible y-values that the function can take. The vertex of the parabola is at (8, -2), and since the parabola opens downward, the maximum y-value is -2.
Step 4: Express the range using interval notation. Since the parabola opens downward and extends infinitely downward, the range includes all y-values less than or equal to -2, which is \((-\infty, -2]\).
Step 5: Summarize the findings: The relation is a function, the domain is all real numbers \((-\infty, \infty)\), and the range is \((-\infty, -2]\).

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Key Concepts

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

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means no vertical line intersects the graph more than once, ensuring each x has a unique y. The vertical line test is a common method to verify if a graph represents a function.
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Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For the given quadratic graph, the domain includes all x-values where the graph exists, typically all real numbers unless restricted by the graph's endpoints or breaks.
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Range of a Function

The range is the set of all possible output values (y-values) that the function can produce. For the parabola shown, the range includes all y-values from the vertex's minimum or maximum point (here, the vertex at (8, -2)) extending upwards or downwards depending on the parabola's direction.
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Related Practice
Textbook Question

Graph each line. Give the domain and range. -3x + 6 = 0

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Textbook Question

Graph each piecewise-defined function.

f(x)={−2xif x<−33x−1if −3≤x≤2−4xif x>2f(x) =\(\begin{cases}\)-2x & \(\text{if }\) x < -3 \\3x - 1 & \(\text{if }\) -3 \(\leq\) x \(\leq\) 2 \\-4x & \(\text{if }\) x > 2\(\end{cases}\)

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Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=-3|x|

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Textbook Question

Graph each function. See Examples 1 and 2. h(x)=|-(1/2)x|

Textbook Question

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

Textbook Question

Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. m=5, b=15