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) = x³ +2
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 17
Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (7/3, 1/5) and (1/3, 6/5)
Verified step by step guidance1
Identify the coordinates of the two points: \(\left(\frac{7}{3}, \frac{1}{5}\right)\) and \(\left(\frac{1}{3}, \frac{6}{5}\right)\).
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 formula:
\[d = \sqrt{\left(\frac{1}{3} - \frac{7}{3}\right)^2 + \left(\frac{6}{5} - \frac{1}{5}\right)^2}\]
Simplify the differences inside the parentheses:
\[\left(\frac{1}{3} - \frac{7}{3}\right) = \frac{1 - 7}{3} = \frac{-6}{3}\]
and
\[\left(\frac{6}{5} - \frac{1}{5}\right) = \frac{6 - 1}{5} = \frac{5}{5}\]
Square each simplified difference, add them, and then take the square root to find the distance. Express the result in simplified radical form before rounding to two decimal places.
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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 a coordinate plane. It is derived from the Pythagorean theorem and given by d = √((x2 - x1)² + (y2 - y1)²). This formula helps find the straight-line distance between points (x1, y1) and (x2, y2).
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Simplifying Radical Expressions
Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares. This makes the answer cleaner and easier to interpret before rounding. For example, √50 can be simplified to 5√2 by factoring 50 into 25 × 2.
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Rounding Decimal Numbers
Rounding is the process of approximating a number to a specified decimal place for simplicity. In this problem, answers are rounded to two decimal places, meaning the number is truncated or increased based on the third decimal digit. This makes results easier to read and use.
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