Textbook Question
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 15
Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (3√3, √5) and (−√3, 4√5)
Verified step by step guidance1
Identify the coordinates of the two points: Point 1 is \(\left(3\sqrt{3}, \sqrt{5}\right)\) and Point 2 is \(\left(-\sqrt{3}, 4\sqrt{5}\right)\).
Recall the distance formula between two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\):
\(d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}\).
Substitute the given coordinates into the distance formula:
\(d = \sqrt{\left(-\sqrt{3} - 3\sqrt{3}\right)^2 + \left(4\sqrt{5} - \sqrt{5}\right)^2}\).
Simplify inside the parentheses:
\(x\)-difference: \(-\sqrt{3} - 3\sqrt{3} = -4\sqrt{3}\)
\(y\)-difference: \(4\sqrt{5} - \sqrt{5} = 3\sqrt{5}\).
Square each difference and add them under the square root:
\(d = \sqrt{\left(-4\sqrt{3}\right)^2 + \left(3\sqrt{5}\right)^2} = \sqrt{16 \times 3 + 9 \times 5}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Distance Formula
The distance formula calculates the distance between two points in the coordinate plane using their coordinates. It is derived from the Pythagorean theorem and is given by d = √((x2 - x1)² + (y2 - y1)²). This formula helps find the straight-line distance between any two points.
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Simplifying Radicals
Simplifying radicals involves expressing square roots in their simplest form by factoring out perfect squares. This process makes the expression easier to interpret and compare. For example, √12 can be simplified to 2√3 by factoring 12 into 4 × 3.
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Rounding Decimal Values
Rounding is the process of approximating a number to a specified degree of accuracy, such as two decimal places. After calculating an exact value, rounding makes the result more practical for interpretation and communication, especially when dealing with irrational numbers.
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Related Practice
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Use the graph of y = f(x) to graph each function g.
g(x) = −ƒ( x/2) +1
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Find the average rate of change of the function from x1 to x2. f(x) = x² + 2x from x1 = 3 to x2 = 5
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The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = 2x + 3
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In Exercises 1-16, use the graph of y = f(x) to graph each function g.
g(x) = -f(2x) - 1
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Textbook Question
Find the domain of each function. f(x) = 1/[4/(x - 1) - 2]
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