Math Problem Statement
Solution
To solve this problem, we are asked to find the composition of two functions and , given by:
We need to find , which is the composition of and . This means that we will substitute into .
Step-by-Step Solution:
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Substitute into :
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Substitute for in :
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Simplify the expression inside the denominator:
So, the expression becomes:
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Final simplified result:
Domain of :
To find the domain, we need to determine where the expression is defined. There are two cases to consider:
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From : Since , the function is undefined when (division by zero).
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From : The expression is undefined when the denominator , i.e., when .
Thus, the domain is all real numbers except and .
Domain in interval notation:
The domain is:
Final Answer:
Let me know if you'd like further details or clarifications! Here are some related questions you might find interesting:
- How do you determine the domain of a composed function in general?
- Can you explain the difference between domain and range in function composition?
- What happens if we reverse the composition and find ?
- How would the range of be determined?
- What role do asymptotes play in determining the behavior of the function ?
Tip: Always simplify the inner function first before composing with the outer function. This often helps in identifying any restrictions on the domain early on.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Algebra
Domain and Range
Formulas
f(x) = 2 / (x + 3)
g(x) = 7 / x
f ∘ g(x) = f(g(x)) = 2 / (g(x) + 3)
Theorems
Domain of Rational Functions
Composition of Functions
Suitable Grade Level
Grades 10-12
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