Textbook Question
The Greatest and Least Integer Functions
For what values of x is
b. ⌈x⌉ = 0
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 1.1.7b
Functions
In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.
b. <IMAGE>
Verified step by step guidance1
To determine if a graph represents a function of x, we can use the 'Vertical Line Test'. This test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.
Examine the given graph visually or conceptually. Imagine drawing vertical lines (lines parallel to the y-axis) across the entire graph.
For each vertical line you imagine, check if it intersects the graph at more than one point. If any vertical line does intersect the graph at more than one point, then the graph is not a function of x.
If every vertical line intersects the graph at most once, then the graph is a function of x.
Provide a reason based on the Vertical Line Test: If the graph passes the test, state that it is a function of x because no vertical line intersects it more than once. If it fails, state that it is not a function of x because there exists at least one vertical line that intersects it more than once.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of a Function
A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. This means that for every x-value in the domain, there is a unique y-value in the range. Understanding this definition is crucial for determining whether a graph represents a function.
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05:43
Definition of the Definite Integral
Vertical Line Test
The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test provides a visual way to assess the uniqueness of outputs for given inputs.
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05:13Slopes of Tangent Lines
Domain and Range
The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Understanding the domain and range helps in analyzing the behavior of functions and identifying whether a graph meets the criteria of a function based on its visual representation.
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5:10Finding the Domain and Range of a Graph
Related Practice
Textbook Question
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
Textbook Question
Industrial costs A power plant sits next to a river where the river is 800 ft wide. Laying a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs \$180 per foot across the river and \$100 per foot along the land.
<IMAGE>
b. Generate a table of values to determine whether the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.
Textbook Question
Combining Functions
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?
b. f/g
Textbook Question
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
Textbook Question
Find the largest interval on which the given function is increasing.
b. ƒ(x) = (x + 1)⁴
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