Math Problem Statement

what is the answer

Determine the sum of the numeric values of the inverse of \( \frac{5x}{10 + x} \)

Solution

To solve the problem:

We need to find the sum of the numeric values of the inverse of the given function:

f(x)=5x10+1x.f(x) = \frac{5x}{10 + 1x}.

Step 1: Write the inverse of the function

The inverse of a function f(x)f(x) is obtained by swapping xx and yy in the equation and solving for yy: y=5x10+x.y = \frac{5x}{10 + x}.

Swap xx and yy: x=5y10+y.x = \frac{5y}{10 + y}.

Solve for yy:

  1. Multiply through by 10+y10 + y to eliminate the denominator: x(10+y)=5y.x(10 + y) = 5y.

  2. Expand the left-hand side: 10x+xy=5y.10x + xy = 5y.

  3. Rearrange terms to isolate yy: 10x=5yxy.10x = 5y - xy.

10x=y(5x).10x = y(5 - x).

  1. Solve for yy: y=10x5x.y = \frac{10x}{5 - x}.

So, the inverse function is: f1(x)=10x5x.f^{-1}(x) = \frac{10x}{5 - x}.

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 xx for which the inverse is defined.

Domain restriction:

The inverse f1(x)=10x5xf^{-1}(x) = \frac{10x}{5 - x} is undefined when the denominator 5x=05 - x = 0, which implies: x=5.x = 5.

Thus, the domain of the inverse is x5x \neq 5.

Numeric values of f1(x)f^{-1}(x):

The numeric values typically refer to specific input values for xx. If the question is asking about summing the values of f1(x)f^{-1}(x) 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:

  1. How is the inverse function related to solving equations in real-life contexts?
  2. What are the key points to verify the domain and range of inverse functions?
  3. How can this function's behavior be visualized graphically?
  4. What role does the denominator 5x5 - x play in defining the inverse?
  5. 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