Math Problem Statement
Solution
Let's break down and solve this problem step by step.
Given Functions:
We are asked to find the following:
We also need to determine the domain for each function.
Domain
The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero. Both and have the denominator , which can be factored as: Thus, and are the values that make the denominator zero. Therefore, the domain of both and is:
1.
Since and have the same denominator, we can directly add the numerators: The domain of is the same as that of and : .
2.
Similarly, subtracting the numerators: The domain of is also .
3.
Multiplying the two functions: Now, expanding the numerator: Thus: The domain remains .
4.
Dividing by : Here, we must exclude any values that make the denominator zero, i.e., , or . Therefore, the domain is:
Summary:
- , Domain:
- , Domain:
- , Domain:
- , Domain:
Would you like more details or have any questions?
Here are some related questions for further understanding:
- How would the domain change if the denominators were different?
- How would the functions behave at the boundaries of their domain (near and )?
- What would happen if the numerator of one function was a quadratic equation?
- Can you simplify complex rational expressions like further?
- How would adding more terms to the numerator affect the overall function?
Tip: Always simplify the function as much as possible before analyzing its domain to avoid mistakes with undefined values.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Functions
Function Operations
Domain of Functions
Formulas
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f * g)(x) = f(x) * g(x)
(f / g)(x) = f(x) / g(x)
Theorems
Domain of a rational function is all real numbers except where the denominator equals zero
Suitable Grade Level
Grades 10-12
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