Math Problem Statement
Solution
Let's analyze the functions given in the image to determine how many are correctly defined.
Function 1:
- The domain is , meaning must be positive.
- The expression inside the cosine function is , which simplifies to . This expression is defined for (no issues with division by zero).
- The cosine function is always defined for any real input.
Thus, is correctly defined on .
Function 2:
- The domain excludes because of the denominator .
- The expression simplifies as: So, .
- For to be defined, the argument of the logarithm , or .
Thus, the domain should be , but the function is undefined for . Since the domain given is , which includes values less than , is not correctly defined.
Function 3:
- The expression inside the square root is , which is always positive for any real , so the square root is defined for all .
- The domain is , which fits the condition for the square root function.
Thus, is correctly defined.
Conclusion:
- and are correctly defined, while is not.
- Therefore, 2 functions are correctly defined.
The correct answer is 2.
Would you like a detailed explanation of any part of this?
Here are some related questions:
- How do we simplify rational expressions?
- Why do logarithmic functions have domain restrictions?
- What are the general rules for the domain of square root functions?
- How does the cosine function behave for different inputs?
- What are the properties of logarithmic functions?
Tip: Always check the domain of the input inside special functions like logarithms and square roots, as they often introduce restrictions.
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Math Problem Analysis
Mathematical Concepts
Function domains
Cosine function
Logarithmic functions
Square root functions
Algebraic simplifications
Formulas
cos(x) for trigonometric functions
ln(x) for logarithmic functions
sqrt(x) for square roots
Theorems
Rules for domains of logarithmic functions
Square root and real numbers theorem
Suitable Grade Level
Undergraduate level (introductory calculus or precalculus)