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Questions and Answers
To find the domain of a composite function, first find the domain of the ______ function.
To find the domain of a composite function, first find the domain of the ______ function.
inside
The composition of functions is denoted as (f o ______)(x).
The composition of functions is denoted as (f o ______)(x).
g
The final domain of a composite function must include the restrictions of both the inside and the ______ functions.
The final domain of a composite function must include the restrictions of both the inside and the ______ functions.
composed
For the function f(x) = √x, its domain is defined as {x | x ≥ ______}.
For the function f(x) = √x, its domain is defined as {x | x ≥ ______}.
The function h(x) = 1/x has a domain restriction that excludes ______.
The function h(x) = 1/x has a domain restriction that excludes ______.
The domain of the function m(x) = x² is {x | x ∈ ______}.
The domain of the function m(x) = x² is {x | x ∈ ______}.
For the composite function h(o)p (x), the output is expressed as ______ = 1/(x² - 10x + 25).
For the composite function h(o)p (x), the output is expressed as ______ = 1/(x² - 10x + 25).
The expression p(o)n(x) simplifies to ______ = x².
The expression p(o)n(x) simplifies to ______ = x².
Flashcards
Domain of a composite function
Domain of a composite function
The domain of the composite function is the set of all inputs that allow both the inner and outer functions to be defined.
Finding the domain of a composite function
Finding the domain of a composite function
The domain of the composite function is the set of all input values of the inner function that result in an output that is within the domain of the outer function.
Composite function restrictions
Composite function restrictions
The domain of the composed function includes the restrictions of both the inner and outer functions.
Composition of functions (f o g)
Composition of functions (f o g)
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Determining the domain of (f o g)
Determining the domain of (f o g)
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Domain analysis for combined functions
Domain analysis for combined functions
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Order of operations in composition
Order of operations in composition
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Applications of function composition
Applications of function composition
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Study Notes
Composite Function Domain
- Steps to find the domain of a composite function:
- Find the domain of the inner function.
- Find the domain of the composed function (result).
- The final domain must include the restrictions from both the inner and composed functions.
Function Compositions
- f(x) = √x, g(x) = √x, h(x) = 1/x
- f(g(x)) = √√x = √x - Domain: x ≥ 0
- h(f(x)) = 1/√x - Domain: x > 0
- g(f(x)) = √√x = √x - Domain: x ≥ 0
- f(h(x)) = √(1/x) - Domain: x > 0
- h(g(x)) = 1/√x - Domain: x > 0
- n(x) = x + 5
Specific Function Compositions
- p(x) = x² - 10x + 25:
- p(n(x)) = (x + 5)² - 10(x + 5) + 25 = x² - 5 +25- Domain: all real numbers
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