Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
____
𝔂 = -2 + √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 1.1.56
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
h(t) = |t³|
Verified step by step guidance1
To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties. A function f(t) is even if f(-t) = f(t) for all t in the domain, and it is odd if f(-t) = -f(t) for all t in the domain.
Consider the given function h(t) = |t³|. First, let's find h(-t) by substituting -t into the function: h(-t) = |-t³|.
Since the cube of a negative number is negative, we have (-t)³ = -(t³). Therefore, |-t³| = |-1 * t³| = |t³|, because the absolute value negates the negative sign.
Now, compare h(-t) with h(t): h(-t) = |t³| and h(t) = |t³|. Since h(-t) = h(t), the function h(t) is even.
Thus, the function h(t) = |t³| is an even function because it satisfies the condition h(-t) = h(t) for all t in its domain.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even Functions
A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. For example, the function f(x) = x² is even because f(-x) = (-x)² = x².
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Exponential Functions
Odd Functions
A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, as f(-x) = (-x)³ = -x³.
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Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is neither even nor odd because it does not satisfy the conditions for either classification. For instance, h(t) = |t³| results in h(-t) = |-t³| = |t³|, which is equal to h(t), indicating it is even, but the cubic term's sign negation complicates its classification.
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Related Practice
Textbook Question
[Technology Exercise]
a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which
3/(x − 1) < 2/(x + 1)
b. Confirm your findings in part (a) algebraically.
Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = |x| - 2
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Textbook Question
Evaluate cos (11π/12) as cos (π/4 + 2π/3).
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 = 1/|x|
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Textbook Question
Theory and Examples
The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.