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Functions and Graphs
Find the domain of y = (x + 3) / (4 − √(x² − 9)).
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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 1.35
Piecewise-Defined Functions
In Exercises 35 and 36, find the (a) domain and (b) range.
𝔂 = { √ -x, -4 ≤ x ≤ 0
{ √ x, 0 < x ≤ 4
Guida verificata passo dopo passo1
Step 1: Understand the piecewise function given. It consists of two parts: y = √(-x) for -4 ≤ x ≤ 0 and y = √x for 0 < x ≤ 4.
Step 2: Determine the domain of the function. The domain is the set of all x-values for which the function is defined. For y = √(-x), x must be between -4 and 0 inclusive. For y = √x, x must be between 0 and 4 inclusive. Combine these intervals to find the overall domain.
Step 3: Determine the range of the function. The range is the set of all possible y-values. For y = √(-x), as x goes from -4 to 0, y goes from 0 to 2. For y = √x, as x goes from 0 to 4, y goes from 0 to 2. Combine these ranges to find the overall range.
Step 4: Consider the endpoints of the intervals. For y = √(-x), when x = -4, y = √4 = 2, and when x = 0, y = √0 = 0. For y = √x, when x = 0, y = √0 = 0, and when x = 4, y = √4 = 2.
Step 5: Conclude the domain and range. The domain is [-4, 4] and the range is [0, 2]. Ensure that the function is continuous across the intervals and check for any discontinuities at the transition point x = 0.
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Piecewise Functions
Piecewise functions are defined by different expressions based on the input value. In this case, the function has two distinct parts: one for the interval from -4 to 0 and another for the interval from 0 to 4. Understanding how to evaluate these functions within their specified domains is crucial for determining their overall behavior.
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Percorso guidato
05:36
Piecewise Functions
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 given piecewise function, the domain is determined by the intervals specified for each piece, which in this case are -4 to 0 and 0 to 4. Identifying the domain helps in understanding the valid inputs for the function.
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Percorso guidato
5:10Finding 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. To find the range of the piecewise function, one must evaluate the outputs of each piece over its respective domain. This involves calculating the minimum and maximum values of the function within the specified intervals.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
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