Math Problem Statement
Solution
Let's go through each question one by one.
Problem 1
Given: Determine the domain and the range .
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Domain : For to be defined, the expression inside the square root must be non-negative: Therefore, the domain is:
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Range : Since yields values from 0 to infinity, will yield values from 1 to negative infinity as increases. Thus, the range is:
Problem 2
Given: Determine , , , and .
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: The composition .
So, .
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Domain : Since is defined for all real numbers, the domain is:
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: The composition .
So, .
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Domain : For , we need to be defined, which requires . Thus, the domain is:
Problem 3
Given: Determine the domain , range , and sketch the graph of the function.
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Domain : Since is defined for all real values of , the domain is:
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Range : Since is an exponential function with a maximum value of 1 (when ) and approaches 0 as approaches , the range is:
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Graph: The graph of is a bell-shaped curve symmetric about the y-axis, peaking at when and gradually approaching as moves away from zero.
Would you like further details on any of these solutions or help with visualizing the graph?
Related Questions
- How do you determine the range of a square root function?
- Why is the domain for composite functions sometimes more restricted?
- What are typical methods for finding the range of exponential functions?
- How does the square root function affect the domain of compositions?
- What makes different from typical exponential functions?
Tip
When determining the range of functions involving square roots or absolute values, carefully consider how these operations limit the possible output values.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Domain and Range
Exponential Functions
Square Root Function
Formulas
Domain condition for square root functions: for √(x - a), x ≥ a
Range of e^(-x^2): (0, 1]
Composition of functions: (f ∘ g)(x) = f(g(x))
Theorems
Domain and Range in Function Composition
Properties of Exponential Functions
Square Root Function Properties
Suitable Grade Level
Grades 11-12
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