Domanda del libro di testo
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = tan(2x - π)
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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 1.19
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = |x| - 2
Guida verificata passo dopo passo1
Step 1: Identify the function type. The given function is 𝑦 = |𝑥| - 2, which is an absolute value function. Absolute value functions are defined for all real numbers.
Step 2: Determine the domain of the function. Since the absolute value function is defined for all real numbers, the domain of 𝑦 = |𝑥| - 2 is all real numbers, which can be expressed as (-∞, ∞).
Step 3: Analyze the transformation of the function. The function 𝑦 = |𝑥| - 2 is a vertical shift of the basic absolute value function 𝑦 = |𝑥|. The graph of 𝑦 = |𝑥| is shifted 2 units downward.
Step 4: Determine the range of the function. The basic absolute value function 𝑦 = |𝑥| has a range of [0, ∞). After shifting the graph 2 units downward, the range of 𝑦 = |𝑥| - 2 becomes [-2, ∞).
Step 5: Summarize the domain and range. The domain of the function 𝑦 = |𝑥| - 2 is all real numbers (-∞, ∞), and the range is [-2, ∞).
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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 function y = |x| - 2, the absolute value function |x| is defined for all real numbers, meaning the domain is all real numbers, or (-∞, ∞). Understanding the domain is crucial for determining the valid inputs for the function.
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Percorso guidato
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 y = |x| - 2, the minimum value occurs when |x| is zero, resulting in y = -2. As x increases or decreases, y increases without bound. Therefore, the range is [-2, ∞), indicating that y can take any value greater than or equal to -2.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This means |x| is always zero or positive. In the function y = |x| - 2, the absolute value affects the shape of the graph, creating a V-like structure that opens upwards, shifted down by 2 units. Understanding this function is essential for analyzing the overall behavior of the given equation.
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06:37Average Value of a Function
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