Math Problem Statement
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Part 1
For f left parenthesis x right parenthesis equals StartFraction 9 Over x plus 6 EndFraction
and g left parenthesis x right parenthesis equals StartFraction 1 Over x EndFraction
, find
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesis; b. the domain of f circle g
Question content area bottom
Part 1
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
StartFraction 9 x Over 1 plus 6 x EndFraction
(Simplify your answer.)
Part 2
b. What is the domain of f circle g?
The domain is
left parenthesis negative infinity comma negative one sixth right parenthesis union left parenthesis negative one sixth comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
.
(Simplify your answer. Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
Solution
We are given two functions and . We need to find:
- (which means ).
- The domain of .
Part 1: Find
To find , we need to substitute into :
Now, substitute for in :
Simplify the expression:
So,
Part 2: Domain of
To find the domain of , we need to consider the restrictions from both and .
- For , the function is undefined when (since division by zero is not allowed). Therefore, .
- For , the function is undefined when , or .
Now, since , we have to ensure that the denominator , which gives .
Thus, the domain excludes and .
Therefore, the domain of is all real numbers except 0 and , written in interval notation as:
Would you like a more detailed explanation of any part? Here are 5 related questions for practice:
- What is the domain of ?
- Find for and .
- How do you determine the domain of a composition of two functions?
- Can a function have more than one domain restriction?
- Why is excluded from the domain of ?
Tip: Always simplify the composition of functions before determining the domain to avoid unnecessary restrictions.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Rational Functions
Domain of Functions
Formulas
(f ∘ g)(x) = f(g(x))
f(x) = 9/(x + 6)
g(x) = 1/x
Theorems
Domain Restrictions in Rational Functions
Suitable Grade Level
Grades 10-12