Math Problem Statement
what is the domain of the composite function f little circle g f(x) =
Solution
To find the domain of the composite function , where:
we need to follow these steps:
1. Find the domain of .
For , the denominator cannot be zero, so: Thus, the domain of is all real numbers except , or:
2. Find the domain of .
Now, we apply inside , so: For the square root to be defined, the expression inside the square root must be non-negative: Solve this inequality: Multiply both sides by , keeping in mind that the direction of the inequality may change depending on whether is positive or negative:
-
Case 1: (i.e., )
In this case, multiply as usual: Simplifying:
-
Case 2: (i.e., )
Here, the inequality flips when multiplying: Simplifying:
3. Combine the results.
From Case 1, we know , and from Case 2, . Therefore, the solution is that , and we must also exclude .
Conclusion:
The domain of the composite function is .
Would you like more details or have any further questions?
Related questions:
- How do you solve compound inequalities involving fractions?
- What is the domain of the square root function in general?
- How can you graphically represent composite functions?
- How do you handle domains when composing two functions?
- What are some common mistakes when finding domains of composite functions?
Tip:
Always remember to check both the inner and outer functions' domains when dealing with composite functions.
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Math Problem Analysis
Mathematical Concepts
Domain of Functions
Composite Functions
Inequalities
Square Roots
Rational Functions
Formulas
Domain restrictions for square root functions: expression inside the square root must be non-negative
Domain restrictions for rational functions: denominator cannot be zero
Theorems
Properties of Composite Functions
Inequality Multiplication Rule
Suitable Grade Level
Grades 10-12
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