Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 4 sin ( 1 )
x
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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 1.1.5
Functions
In Exercises 1–6, find the domain and range of each function.
f(t) = 4/(3 − t)
Verified step by step guidance1
To find the domain of the function \( f(t) = \frac{4}{3 - t} \), we need to determine the values of \( t \) for which the function is defined. The function is undefined when the denominator is zero.
Set the denominator equal to zero and solve for \( t \): \( 3 - t = 0 \).
Solving \( 3 - t = 0 \) gives \( t = 3 \). Therefore, the function is undefined at \( t = 3 \).
The domain of \( f(t) \) is all real numbers except \( t = 3 \). In interval notation, this is \( (-\infty, 3) \cup (3, \infty) \).
The range of \( f(t) \) is all real numbers except zero, because the function can take any real value except zero as \( t \) approaches 3 from either side, making the function approach positive or negative infinity.
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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 rational functions like f(t) = 4/(3 - t), the domain excludes any values that make the denominator zero, as division by zero is undefined. In this case, t cannot equal 3, so the domain is all real numbers except t = 3.
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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 function f(t) = 4/(3 - t), as t approaches 3, the function approaches infinity or negative infinity, depending on the direction of approach. Thus, the range includes all real numbers except for the value that the function approaches as t nears 3, which is 0.
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Vertical Asymptote
A vertical asymptote is a line that a graph approaches but never touches or crosses, typically occurring where a function is undefined. In the case of f(t) = 4/(3 - t), the vertical asymptote is at t = 3, indicating that as t approaches 3 from either side, the function's value increases or decreases without bound. This concept is crucial for understanding the behavior of the function near its domain restrictions.
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Related Practice
Textbook Question
Graph the functions in Exercises 37–56.
y = (x + 1)²/³
207
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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 9–16, determine whether the function is even, odd, or neither.
𝔂 = x cos x
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Textbook Question
Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
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