BackStep-by-Step Guidance for Composition and Decomposition of Functions
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Q1. Let and . Find:
(a)
(b)
Background
Topic: Composition of Functions
This question tests your understanding of how to compose two functions, meaning how to substitute one function into another.
Key Terms and Formulas
Composition:
Domain: The set of values for which the composition is defined.
Step-by-Step Guidance
For , start by finding , which is .
Substitute into : .
Write out the expression: .
Expand and separately.
Combine like terms to simplify the expression, but stop before the final simplification.
Try solving on your own before revealing the answer!
Final Answer:
We substituted into and simplified the result.
Q2. Let and . Find:
(a)
(b)
(c)
(d) Domain of
Background
Topic: Composition of Functions and Domain Analysis
This question tests your ability to compose functions, evaluate compositions at specific values, and determine the domain of a composite function.
Key Terms and Formulas
Composition:
Domain: Values of for which the function is defined (no division by zero, etc.)
Step-by-Step Guidance
For , first find .
Substitute into : .
Write the expression: .
Simplify the denominator: .
For , substitute into the simplified expression, but stop before calculating the final value.
Try solving on your own before revealing the answer!
Final Answer:
For , substitute to get .
Q3. Let and . Find and state the domain.
Background
Topic: Composition of Functions and Domain Analysis
This question tests your ability to compose a linear function with a square root function and determine the domain of the composite function.
Key Terms and Formulas
Composition:
Domain: For , .
Step-by-Step Guidance
Find .
Substitute into : .
Write the expression: .
Determine the domain: Set and solve for .
Express the domain in interval notation, but stop before stating the exact interval.
Try solving on your own before revealing the answer!
Final Answer:
Domain:
Q4. Decompose the following functions: Find and such that .
(a)
(b)
Background
Topic: Decomposition of Functions
This question tests your ability to break a function into two simpler functions whose composition gives the original function.
Key Terms and Formulas
Decomposition:
Step-by-Step Guidance
For (a), identify an inner function that is inside another operation (e.g., ).
Let , then so that .
For (b), identify the inner function , and the outer function .
Check that matches for each case.
Try solving on your own before revealing the answer!
Final Answer:
(a) ,
(b) ,
