Textbook Question
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = - 3/5, passing through (10, −4)
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 24
In Exercises 11–26, determine whether each equation defines y as a function of x. xy - 5y =1
Verified step by step guidance1
Start with the given equation: \(xy - 5y = 1\).
Factor out \(y\) from the left side: \(y(x - 5) = 1\).
Solve for \(y\) by dividing both sides by \((x - 5)\), assuming \(x \neq 5\): \(y = \frac{1}{x - 5}\).
Analyze the expression for \(y\): for each value of \(x\) (except \(x = 5\)), there is exactly one corresponding value of \(y\).
Conclude that since \(y\) can be written as a single-valued function of \(x\) (except at \(x = 5\) where it is undefined), the equation defines \(y\) as a function of \(x\) on its domain.
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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). To determine if an equation defines y as a function of x, we check if for every x there is only one y that satisfies the equation.
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Graphs of Common Functions
Implicit vs. Explicit Functions
An explicit function expresses y directly in terms of x (e.g., y = f(x)), while an implicit function involves both variables in an equation (e.g., xy - 5y = 1). Understanding how to manipulate implicit equations helps determine if y can be uniquely solved for each x.
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Solving for y and the Vertical Line Test
To check if y is a function of x, solve the equation for y. If you get one unique y for each x, it is a function. Graphically, the vertical line test states that if any vertical line crosses the graph more than once, y is not a function of x.
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Related Practice
Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. |x| − y = 2
Textbook Question
Find the domain of each function. f(x) = √(24 - 2x)
Textbook Question
Use the graph of y = f(x) to graph each function g. g(x) = f(-x)
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Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = f(-x)+1
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Textbook Question
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ is perpendicular to the line whose equation is 3x - 2y - 4 = 0 and has the same y-intercept as this line.