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 average rate of change of the function from x1 to x2. f(x) = √x from x1 = 4 to x2 = 9
Verified step by step guidance1
Identify the function given: \(f(x) = \sqrt{x}\), and the interval endpoints \(x_1 = 4\) and \(x_2 = 9\).
Recall the formula for the average rate of change of a function \(f(x)\) over the interval \([x_1, x_2]\):
\(\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\).
Calculate the function values at the endpoints:
\(f(x_1) = f(4) = \sqrt{4}\) and \(f(x_2) = f(9) = \sqrt{9}\).
Substitute these values into the average rate of change formula:
\(\frac{\sqrt{9} - \sqrt{4}}{9 - 4}\).
Simplify the numerator and denominator separately, then divide to find the average rate of change over the interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Average Rate of Change
The average rate of change of a function between two points measures how much the function's output changes per unit change in input. It is calculated as the difference in function values divided by the difference in input values, similar to the slope of a secant line connecting the two points.
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Change of Base Property
Square Root Function
The square root function, denoted as f(x) = √x, outputs the non-negative number whose square is x. It is defined for x ≥ 0 and is a common example of a radical function, which often requires careful evaluation when calculating function values at specific points.
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Evaluating Functions at Given Points
To find the average rate of change, you must correctly evaluate the function at the specified input values. This involves substituting the input values into the function and simplifying, ensuring accurate calculation of the function's outputs at those points.
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Related Practice
Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. x = y²
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
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = -5, passing through (-4, -2)
Textbook Question
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 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)
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