Math Problem Statement
f(x) = 1/x g(x) = (x^2 - 2x + 3)/(x - 3) Find the domain of the composite function f(g(x))
Solution
To find the domain of the composite function , we need to determine the values of for which the composition is defined.
Steps:
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Domain of : The function is undefined when the denominator is 0, i.e., when: So, the domain of is all real numbers except .
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Domain of : The function is undefined when , so for to be defined, .
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Solve : To ensure that is defined, we need to find when : The quadratic equation has no real solutions because its discriminant is negative: Since there are no real solutions to this equation, never equals 0.
Conclusion:
The domain of is the same as the domain of , except for when , which doesn't happen for any real . Therefore, the only restriction is that .
Thus, the domain of is:
Would you like a more detailed explanation of any step?
Here are 5 related questions you might find useful:
- How do you find the domain of the sum of two functions?
- What happens to the domain if the composite function has a square root?
- How do you find the range of a composite function?
- What is the difference between a composite function and a product of functions?
- Can the range of affect its domain?
Tip: Always start by finding the domain of the inner function before considering the outer function when working with composite functions.
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Math Problem Analysis
Mathematical Concepts
Composite Functions
Domain of a Function
Rational Functions
Quadratic Equations
Formulas
f(x) = 1/x
g(x) = (x^2 - 2x + 3)/(x - 3)
Theorems
Domain of a composite function
Discriminant of a quadratic equation
Suitable Grade Level
Grades 9-11
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