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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 1.2.43
Chapter 1, Problem 1.2.43

Graph the functions in Exercises 37–56.


y = (x + 1)²/³

Verified step by step guidance
1
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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Introduction 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

Solving Trigonometric Equations


For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.


sin² θ = cos² θ

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Textbook Question

Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


sin x²

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Textbook Question

Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = −x³

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Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph the function f (x) = sin 2x + cos 3x.

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Textbook Question

Using the Addition Formulas


Use the addition formulas to derive the identities in Exercises 31–36.


sin (A − B) = sin A cos B − cos A sin B

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Textbook Question

General Sine Curves


For


f(x) = A sin ((2π/B)(x – C) +D


identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.


y = ½ sin (πx – x) + ½

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