https://file.aiquickdraw.com/mathbot/file/1726627107767h49daff0.jpg

Let's go over the three problems from the image you've uploaded.

### Problem 40:
\[
f(x) + 3 \cdot f\left( \frac{1}{x} \right) = \frac{1}{x}
\]

This is a functional equation. The goal is to find \( f(x) \). We can try some specific values of \( x \) to simplify the equation and look for a pattern.

### Problem 41:
\[
f(x+3) = x^2 - 4
\]

Here, you are given the value of \( f(x+3) \) as a quadratic function in \( x \). The task is to find \( f(x) \) by substituting \( x \) values and possibly shifting the function.

### Problem 42:
\[
2f(x) + f\left( \frac{1}{x} \right) = x
\]

Again, this is a functional equation where the goal is to find \( f(x) \) by solving the equation for different values of \( x \).

Would you like me to proceed with solving these step by step? Let me know which one you'd like to start with!

40. f(x) + 3 * f(1/x) = 1/x
41. f(x+3) = x^2 - 4
42. 2f(x) + f(1/x) = x

Explore the solutions to functional equations such as f(x) + 3 * f(1/x) = 1/x and f(x+3) = x^2 - 4. Understand key algebraic concepts and properties of quadratic functions. Suitable for high school students in Grades 10-12.

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