Textbook Question
Functions and Graphs
Find the natural domain and graph the functions in Exercises 15–20.
f(x) = 1 − 2x − x²
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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.P.31
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 4 sin ( 1 )
x
Verified step by step guidance1
Step 1: Identify the function type. The given function is y = 4 sin(1)x, which is a trigonometric function involving sine.
Step 2: Determine the domain of the function. The sine function, sin(x), is defined for all real numbers. Therefore, the domain of y = 4 sin(1)x is all real numbers, which can be expressed as (-∞, ∞).
Step 3: Understand the range of the sine function. The basic sine function, sin(x), has a range of [-1, 1].
Step 4: Consider the effect of the coefficient on the range. The function y = 4 sin(1)x involves a multiplication by 4, which stretches the range of the sine function. Therefore, the range of y = 4 sin(1)x is [-4, 4].
Step 5: Summarize the findings. The domain of the function is all real numbers, and the range is [-4, 4].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For trigonometric functions like sine, the domain is typically all real numbers unless restricted by other factors, such as division by zero or square roots of negative numbers.
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Range of a Function
The range of a function is the set of all possible output values (y-values) that the function can produce. For the sine function, the range is limited to values between -1 and 1. When multiplied by a constant, such as 4 in this case, the range adjusts accordingly to between -4 and 4.
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Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The sine function specifically measures the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is crucial for analyzing periodic behavior and determining the domain and range of expressions involving them.
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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)
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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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Textbook Question
Theory and Examples
The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.
a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)
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Textbook Question
Functions and Graphs
Graph the following equations and explain why they are not graphs of functions of x.
a. |y| = x
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