Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

2. Graphs of Equations

Graphs and Coordinates

College Algebra

2. Graphs of Equations

Graphs and Coordinates

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0:45 minutes

Problem 58

Textbook Question

In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.

Verified step by step guidance

Step 1: Understand the vertical line test. The vertical line test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Step 2: Observe the given graph. Notice the red curve plotted on the coordinate plane.

Step 3: Imagine drawing vertical lines at various points along the x-axis. For example, draw vertical lines at x = -5, x = 0, and x = 5.

Step 4: Check the intersections of these vertical lines with the graph. Each vertical line should intersect the graph at exactly one point.

Step 5: Conclude that since no vertical line intersects the graph at more than one point, the graph passes the vertical line test and y is 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 intersects the graph at more than one point, the graph does not represent a function, as this indicates that a single input (x-value) corresponds to multiple outputs (y-values). This test is essential for understanding the relationship between variables in a function.

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. This means that for every x-value in the domain, there is a unique y-value in the range. Understanding this definition is crucial for identifying functions and applying the vertical line test effectively.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of mathematical relationships. In the context of functions, it requires understanding how the shape and behavior of a graph reflect the relationship between x and y. This skill is vital for applying the vertical line test and determining whether a graph represents a function.