Textbook Question
For the pair of functions defined, find (ƒ+g)(x). Give the domain of each. See Example 2.
ƒ(x)=√(4x-1), g(x)=1/x
Ch. 2 - Graphs and Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 3, Problem 23
Determine whether the three points are the vertices of a right triangle. (-6,-4),(0,-2),(-10,8)
Verified step by step guidance1
Identify the three points given: A(-6, -4), B(0, -2), and C(-10, 8).
Calculate the distance between each pair of points using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
Find the lengths of the sides: \(AB\), \(BC\), and \(AC\) by substituting the coordinates of the points into the distance formula.
Check if the triangle is right-angled by verifying the Pythagorean theorem: see if the square of the longest side equals the sum of the squares of the other two sides, i.e., \(c^2 = a^2 + b^2\).
Conclude whether the points form a right triangle based on the Pythagorean theorem check.
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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 length between two points in the coordinate plane using their coordinates. It is derived from the Pythagorean theorem and is given by √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the lengths of the sides of the triangle formed by the given points.
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Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse length equals the sum of the squares of the other two sides. To determine if a triangle is right-angled, check if the squares of the two shorter sides add up to the square of the longest side.
Identifying Triangle Vertices and Side Lengths
To analyze a triangle from points, first identify all three vertices and calculate the lengths of all sides using the distance formula. Comparing these lengths allows you to apply the Pythagorean theorem and verify if the triangle is right-angled.
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Related Practice
Textbook Question
Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. vertical, through (-6, 4)
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Textbook Question
For each graph, determine whether y is a function of x. Give the domain and range of each relation.
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Textbook Question
Graph each function. See Examples 1 and 2. g(x)=(1/2)x2
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Textbook Question
Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.
Textbook Question
Graph each piecewise-defined function.
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