Textbook Question
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31β36.
sin (A β B) = sin A cos B β cos A sin B
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.28
In Exercises 19β32, find the (a) domain and (b) range.
_____
π = -1 + β 2 - x
Verified step by step guidance1
Step 1: Identify the function type. The given function is π¦ = -1 + β(2 - x), which involves a cube root. Cube root functions are defined for all real numbers, so there are no restrictions on the domain.
Step 2: Determine the domain. Since the cube root function is defined for all real numbers, the domain of π¦ = -1 + β(2 - x) is all real numbers, or (-β, β).
Step 3: Analyze the range of the function. The cube root function, β(x), can take any real number as its output, ranging from -β to β.
Step 4: Consider the transformations applied to the cube root function. The function π¦ = -1 + β(2 - x) involves a vertical shift downward by 1 unit. This does not affect the range of the cube root function, which remains all real numbers.
Step 5: Conclude the range. Since the cube root function can output any real number and the vertical shift does not restrict this, the range of the function is also all real numbers, or (-β, β).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the given function π = -1 + β(2 - x), we need to determine the values of x that do not lead to undefined expressions. Since the cube root function is defined for all real numbers, the domain is all real numbers where 2 - x is a real number, which translates to x being less than or equal to 2.
Recommended video:
5:10
Finding the Domain and Range of a Graph
Range
The range of a function is the set of all possible output values (y-values) that the function can produce. In the case of π = -1 + β(2 - x), as x approaches negative infinity, the cube root term will also approach negative infinity, and as x approaches 2, the output approaches -1. Therefore, the range of this function is all real numbers greater than or equal to -1.
Recommended video:
5:10Finding the Domain and Range of a Graph
Cube Root Function
The cube root function, denoted as βx, is a function that returns the number which, when cubed, gives the input x. This function is defined for all real numbers, meaning it can take any real number as input and will produce a real number as output. Understanding the properties of the cube root function is essential for analyzing the behavior of the given function, particularly in determining its domain and range.
Recommended video:
5:57Graphs of Common Functions
Related Practice
Textbook Question
In Exercises 5β8, determine whether the graph of the function is symmetric about the π-axis, the origin, or neither.
π = eβ»Λ£Β²
1
views
Textbook Question
Even and Odd Functions
In Exercises 47β62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = xΒ² + x
1
views
Textbook Question
In Exercises 9β16, determine whether the function is even, odd, or neither.
π = x - sin x
1
views
Textbook Question
Use graphing software to graph the functions specified in Exercises 31β36.
Select a viewing window that reveals the key features of the function.
Graph two periods of the function f (x) = 3 cot (x/2) + 1.
Textbook Question
Even and Odd Functions
In Exercises 47β62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = xβ»β΅