Domanda del libro di testo
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
sin (2π − x)
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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 1.27
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = cos(x - 3) + 1
Guida verificata passo dopo passo1
Step 1: Understand the function y = cos(x - 3) + 1. This is a transformation of the basic cosine function, where the graph is shifted horizontally by 3 units to the right and vertically by 1 unit upwards.
Step 2: Determine the domain of the function. The cosine function, cos(x), is defined for all real numbers. Therefore, the domain of y = cos(x - 3) + 1 is also all real numbers, which can be expressed as (-∞, ∞).
Step 3: Analyze the range of the function. The basic cosine function, cos(x), has a range of [-1, 1]. The transformation y = cos(x - 3) + 1 shifts the entire range up by 1 unit.
Step 4: Calculate the new range. By shifting the range [-1, 1] up by 1 unit, the new range becomes [0, 2]. This is because the minimum value -1 becomes 0 and the maximum value 1 becomes 2.
Step 5: Summarize the findings. The domain of the function y = cos(x - 3) + 1 is (-∞, ∞), and the range is [0, 2].
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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 the function y = cos(x - 3) + 1, the cosine function is defined for all real numbers, so the domain is all real numbers, denoted as (-∞, ∞).
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Percorso guidato
5:10
Finding the Domain and Range of a Graph
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 y = cos(x - 3) + 1, the cosine function oscillates between -1 and 1, so when shifted up by 1, the range becomes [0, 2].
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Percorso guidato
5:10Finding the Domain and Range of a Graph
Transformation of Functions
Transformations involve shifting, stretching, or compressing the graph of a function. In y = cos(x - 3) + 1, the graph of cos(x) is horizontally shifted right by 3 units and vertically shifted up by 1 unit, affecting the range but not the domain.
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Percorso guidato
5:25Intro to Transformations
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In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x⁴ + 1
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