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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 21
Chapter 3, Problem 21

For each graph, determine whether y is a function of x. Give the domain and range of each relation.
Graph showing two parabolas, one opening upward above y=0 and one opening downward below y=0, on an xy-plane.

Verified step by step guidance
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Step 1: Identify if y is a function of x by using the vertical line test. A relation is a function if and only if no vertical line intersects the graph at more than one point.
Step 2: Observe the graph. Notice that for some values of x, there are two corresponding y-values (for example, at x = 0, y = 11 and y = -11). This means the vertical line at x = 0 intersects the graph at two points.
Step 3: Since the vertical line at x = 0 intersects the graph at two points, y is not a function of x for this relation.
Step 4: Determine the domain by looking at the x-values covered by the graph. The graph extends horizontally from about -20 to 20, so the domain is approximately \([-20, 20]\).
Step 5: Determine the range by looking at the y-values covered by the graph. The graph extends vertically from about -20 to 20, so the range is approximately \([-20, 20]\).

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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 at more than one point. Understanding this helps determine if y is a function of x by checking if any x-value maps to multiple y-values.
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Domain and Range

The domain of a relation is the set of all possible x-values, while the range is the set of all possible y-values. Identifying these sets from a graph involves looking at the horizontal spread for the domain and the vertical spread for the range.
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Vertical Line Test

The vertical line test is a visual method to determine if a graph represents a function. If any vertical line drawn through the graph intersects it more than once, the graph does not represent a function. This test is essential for analyzing the given parabolic curves.
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Related Practice
Textbook Question

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

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Textbook Question

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. P(6,-2), Q(4,6)

Textbook Question

Graph each piecewise-defined function.

f(x)={x−1if x≤32if x>3f(x) =\(\begin{cases}\)x - 1 & \(\text{if }\) x \(\leq\) 3 \\2 & \(\text{if }\) x > 3\(\end{cases}\)

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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.

x-intercept (3,0), y-intercept (0,-2)

Textbook Question

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

ƒ(x)=2x^2-3x, g(x)=x^2-x+3

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. center (√2, √2), radius √2

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