Textbook Question
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 59
Use the vertical line test to identify graphs in which y is a function of x.
Verified step by step guidance1
Recall that the vertical line test is used to determine if a graph represents a function. According to this test, if any vertical line intersects the graph at more than one point, then the graph does not represent y as a function of x.
Observe the given graph, which is an oval shape (an ellipse) centered roughly around (-5, 5) on the coordinate plane.
Imagine drawing vertical lines (lines parallel to the y-axis) at various x-values across the graph. Notice that for many x-values within the range of the oval, the vertical line will intersect the oval at two points: one on the upper half and one on the lower half of the oval.
Since there are vertical lines that intersect the graph at more than one point, this means that for some x-values, there are multiple corresponding y-values.
Therefore, based on the vertical line test, the graph does not represent y as a function of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Line Test
The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function because a single input (x-value) corresponds to multiple outputs (y-values).
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Definition of a Function
A function is a relation where each input (x-value) has exactly one output (y-value). This means no x-value can be paired with more than one y-value. Understanding this definition is essential to applying the vertical line test correctly.
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Graph Interpretation
Interpreting graphs involves understanding the relationship between x and y coordinates. In the given image, the oval shape crosses some vertical lines more than once, indicating multiple y-values for a single x-value, which helps conclude whether the graph is a function.
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Related Practice
Textbook Question
Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² - 2x + y² – 15 = 0
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Textbook Question
Use the vertical line test to identify graphs in which y is a function of x.
Textbook Question
Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. (fog) (0)
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Textbook Question
Find a. (fog) (x) b. (go f) (x) f(x) = 4-x, g(x) = 2x² +x+5
Textbook Question
Find a. (fog) (2) b. (go f) (2) f(x) = 4-x, g(x) = 2x² +x+5