Math Problem Statement
Solution
We are given the following functions:
We are tasked with finding , which represents , and determining its domain.
Step 1: Evaluate
First, substitute into :
Step 2: Evaluate
Next, substitute into : So, .
Step 3: Determine the Domain
The domain is determined by the conditions that must be satisfied for the expression to be defined:
- For the square root function , we require , which simplifies to .
- The denominator cannot be zero because division by zero is undefined. Solving , we get , which must be excluded from the domain.
Thus, the domain is .
Final Answer:
with the domain .
Would you like more details or have any questions?
Here are 5 related questions to further explore the topic:
- How would the domain change if were defined as instead of ?
- What is the range of ?
- How would you calculate the inverse of ?
- How would you graph and indicate its domain visually?
- How does composition of functions affect the domain of the resulting function?
Tip: Always check both the inner and outer functions for domain restrictions when dealing with composite functions.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Domain of Functions
Radicals
Algebra
Formulas
(f o h o g)(x) = f(h(g(x)))
Square root function: sqrt(x)
Theorems
Domain Restrictions for Radical Functions
Composite Function Definition
Suitable Grade Level
Grades 10-12
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