https://file.aiquickdraw.com/mathbot/file/1725983351826sima2lft.jpg

The problem in the image asks to find the function \( g(x) \), given two equations:

1. \( f(x) = \frac{1}{x} \)
2. \( \left( \frac{f}{g} \right)(x) = \frac{x + 1}{x^2 - x} \)

We need to solve for \( g(x) \).

### Solution:

The expression \( \left( \frac{f}{g} \right)(x) \) implies:
\[
\frac{f(x)}{g(x)} = \frac{x + 1}{x^2 - x}
\]

Substitute \( f(x) = \frac{1}{x} \) into the equation:
\[
\frac{\frac{1}{x}}{g(x)} = \frac{x + 1}{x^2 - x}
\]

Simplify the left-hand side:
\[
\frac{1}{x g(x)} = \frac{x + 1}{x(x - 1)}
\]

Now, multiply both sides of the equation by \( x \):
\[
\frac{1}{g(x)} = \frac{x + 1}{x - 1}
\]

To isolate \( g(x) \), take the reciprocal of both sides:
\[
g(x) = \frac{x - 1}{x + 1}
\]

Thus, the function \( g(x) \) is:
\[
g(x) = \frac{x - 1}{x + 1}
\]

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Would you like further details on any step, or do you have additional questions?

Here are 5 questions to expand on this topic:

1. What is the meaning of taking the reciprocal of both sides in an equation?
2. How can this process be applied to more complex functions of \( f(x) \) and \( g(x) \)?
3. What are the restrictions on \( x \) for which this solution is valid?
4. How would you solve this if the functions were trigonometric instead of algebraic?
5. Can this method be generalized to solve similar types of fraction-based function equations?

**Tip:** Always check the domain of your final answer by considering any values of \( x \) that might cause division by zero or other undefined operations.

Given f(x) = 1/x and (f/g)(x) = (x + 1)/(x^2 - x), find the function g.

Learn how to find the function g(x) given f(x) = 1/x and (f/g)(x) = (x + 1)/(x^2 - x). This problem covers key algebraic concepts and function manipulation, suitable for students in Grades 10-12.

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