Domanda del libro di testo
[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.
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 1, Problema 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|
Guida verificata passo dopo passo1
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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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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6:13
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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Percorso guidato
06:21Properties of Functions
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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Percorso guidato
06:37Average Value of a Function
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