Math Problem Statement
Find domain and range
Solution
The expression in the image is:
Finding the Domain
To find the domain of the function, we need to determine for which values of and the expression is defined.
- The expression implies that the expression is defined for all and such that (since for any real number ).
- Therefore, .
Thus, the domain of the function is all real numbers except :
Finding the Range
To find the range, we solve the equation for :
Since , can take any non-zero real value. Therefore, can also take any non-zero real value.
- Since , can be any non-zero real number.
- itself can therefore be any non-zero real number.
Thus, the range of the function is all real numbers except :
Would you like more details, or do you have any questions?
Related Questions:
- What happens to the graph of the function as approaches 9?
- Can we rewrite this expression in terms of inverse functions?
- How would you describe the behavior of as becomes very large or very small?
- What are the asymptotes for this function, if any?
- How would you plot this function on a coordinate plane?
Tip:
When determining the domain and range of a function, always consider the values that make the function undefined, such as division by zero or taking the square root of a negative number in the real number system.
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Math Problem Analysis
Mathematical Concepts
Functions
Domain and Range
Formulas
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Theorems
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Suitable Grade Level
High School