Math Problem Statement
what is the answer
Solution
To solve the problem:
We need to find the sum of the numeric values of the inverse of the given function:
Step 1: Write the inverse of the function
The inverse of a function is obtained by swapping and in the equation and solving for :
Swap and :
Solve for :
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Multiply through by to eliminate the denominator:
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Expand the left-hand side:
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Rearrange terms to isolate :
- Solve for :
So, the inverse function is:
Step 2: Determine the numeric values of the inverse
The problem asks for the sum of the numeric values of the inverse. To do this, we must find the values of for which the inverse is defined.
Domain restriction:
The inverse is undefined when the denominator , which implies:
Thus, the domain of the inverse is .
Numeric values of :
The numeric values typically refer to specific input values for . If the question is asking about summing the values of at some predefined points, please clarify or provide additional details.
Alternatively, if it refers to summing all valid numeric outputs (which is unusual in context), the sum is undefined due to the continuous range.
Would you like to clarify the input, or should we assume specific points for summation?
Expanded Questions:
- How is the inverse function related to solving equations in real-life contexts?
- What are the key points to verify the domain and range of inverse functions?
- How can this function's behavior be visualized graphically?
- What role does the denominator play in defining the inverse?
- How do restrictions in domain influence real-world applications of functions?
Tip:
When solving inverse functions, always ensure you check for any restrictions in the domain and range caused by division by zero or square roots of negative numbers!
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Algebraic Manipulation
Domain and Range Analysis
Formulas
Inverse of a function: Swap x and y and solve for y
y = \frac{10x}{5 - x} (inverse of the given function)
Theorems
Inverse Function Theorem
Suitable Grade Level
Grades 10-12