Textbook Question
Determine whether the three points are the vertices of a right triangle. See Example 3.
Ch. 2 - Graphs and Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 3, Problem 25
Determine whether each equation defines y as a function of x. y = 6 -x2
Verified step by step guidance1
Recall the definition of a function: for each input value of x, there must be exactly one output value of y.
Look at the given equation: \(y = 6 - x^{2}\).
For any value of x, calculate the right side \(6 - x^{2}\) to find the corresponding y value.
Since squaring x (\(x^{2}\)) always gives a non-negative result and subtracting it from 6 gives a unique y for each x, there is only one y for each x.
Therefore, the equation defines y as a function of x because each x corresponds to exactly one y.
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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 for every x, there is only one y. Understanding this helps determine if an equation defines y as a function of x.
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Graphs of Common Functions
Evaluating Equations for Function Criteria
To check if an equation defines y as a function of x, substitute values of x and see if y has a unique value. If each x produces only one y, the equation represents a function.
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Quadratic Functions and Their Graphs
The equation y = 6 - x² is a quadratic function, which graphs as a parabola opening downward. Since for each x there is only one y, quadratic equations like this define y as a function of x.
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5:26Graphs of Logarithmic Functions
Related Practice
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