Textbook Question
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 a. f(6)
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 27
Find the midpoint of each line segment with the given endpoints. (8, 3√5) and (−6, 7√5)
Verified step by step guidance1
Recall that the midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{ x_1 + x_2 }{ 2 }, \frac{ y_1 + y_2 }{ 2 } \right) \]
Identify the coordinates of the given endpoints:
\( (x_1, y_1) = (8, 3\sqrt{5}) \) and \( (x_2, y_2) = (-6, 7\sqrt{5}) \).
Calculate the midpoint's \( x \)-coordinate by adding the \( x \)-values and dividing by 2:
\[ \frac{8 + (-6)}{2} \]
Calculate the midpoint's \( y \)-coordinate by adding the \( y \)-values and dividing by 2:
\[ \frac{3\sqrt{5} + 7\sqrt{5}}{2} \]
Combine the results to write the midpoint as:
\[ \left( \frac{8 + (-6)}{2}, \frac{3\sqrt{5} + 7\sqrt{5}}{2} \right) \]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Formula
The midpoint formula calculates the point exactly halfway between two given points in a coordinate plane. It is found by averaging the x-coordinates and the y-coordinates separately: Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). This formula helps locate the center of a line segment.
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Coordinate Geometry
Coordinate geometry involves representing geometric figures using coordinates on the Cartesian plane. Understanding how points, lines, and shapes relate through their coordinates is essential for applying formulas like the midpoint formula and interpreting results accurately.
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Simplifying Radicals
Simplifying radicals means expressing square roots in their simplest form by factoring out perfect squares. This skill is important when working with coordinates involving square roots, ensuring the final answer is presented clearly and correctly.
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Related Practice
Textbook Question
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 c. f(-x)
Textbook Question
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 +1)/(x-3)
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Textbook Question
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 b. f(x + 1)
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Determine whether each equation defines y as a function of x. |x|- y = 5
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Textbook Question
Find the domain of each function. g(x) = √(x −2) /(x-5)
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