Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 4 sin ( 1 )
x
293
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.2.43
Graph the functions in Exercises 37–56.
y = (x + 1)²/³
Verified step by step guidance1
Identify the type of function: The given function is y = (x + 1)^(2/3). This is a power function with a fractional exponent, which indicates a root function.
Determine the domain: Since the base of the power (x + 1) is raised to a fractional power with an even numerator, the expression is defined for all real numbers x. Therefore, the domain is all real numbers.
Find the critical points: To find critical points, take the derivative of the function with respect to x. Use the chain rule to differentiate y = (x + 1)^(2/3).
Analyze the behavior at critical points: Evaluate the derivative to find where it is zero or undefined. These points will help determine local maxima, minima, or points of inflection.
Sketch the graph: Use the information from the derivative and critical points to sketch the graph. Consider the symmetry, intercepts, and end behavior of the function to complete the graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a function. It requires understanding the function's domain, range, and any transformations applied to the basic function. For y = (x + 1)²/³, identifying key features like intercepts, asymptotes, and symmetry helps in sketching an accurate graph.
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Graph of Sine and Cosine Function
Fractional Exponents
Fractional exponents, such as ²/³, indicate roots and powers. The expression (x + 1)²/³ can be interpreted as the cube root of (x + 1) squared. Understanding fractional exponents is crucial for determining the function's behavior, especially near critical points where the base might be zero or negative.
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Guided course
7:39Introduction to Exponent Rules
Transformations of Functions
Transformations involve shifting, stretching, or compressing the graph of a function. The expression y = (x + 1)²/³ includes a horizontal shift left by 1 unit due to the (x + 1) term. Recognizing transformations helps in predicting how the graph will move or change shape compared to the parent function y = x²/³.
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Related Practice
Textbook Question
Functions
In Exercises 1–6, find the domain and range of each function.
f(t) = 4/(3 − t)
416
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Textbook Question
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
cos (3π/2 + x)
240
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Textbook Question
Finding Formulas for Functions
Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.
356
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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
________
𝔂 = 5 - √ x² - 2x - 3
334
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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 2 + 3x² .
x² + 4
250
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