To find the inverse of the function $f(x) = 5x + 3$ using the method of solving algebraically (also known as the "rubic" method), follow these steps:

Steps to Find the Inverse:

1. Write the function: $y = 5x + 3$ Here, $y$ represents the output of the function $f(x)$.

2. Switch $x$ and $y$: To find the inverse, swap $x$ and $y$: $x = 5y + 3$

3. Solve for $y$: Now, solve for $y$ in terms of $x$.

   • Subtract 3 from both sides: $x - 3 = 5y$

   • Divide by 5: $y = \frac{x - 3}{5}$

4. Rewrite the inverse function: Now, the inverse function is: $f^{-1}(x) = \frac{x - 3}{5}$

Final Answer:

The inverse function of $f(x) = 5x + 3$ is: $f^{-1}(x) = \frac{x - 3}{5}$

Would you like further details or clarifications on any step?

Here are 5 additional questions based on the above process:

1. How can you check if a function and its inverse are correct?
2. What are the conditions for a function to have an inverse?
3. What happens if you apply both the function and its inverse in succession?
4. Can you describe a real-world situation where finding an inverse function is useful?
5. How would the process change if the function was $f(x) = 5x^2 + 3$?

Tip: To verify that you've correctly found the inverse, apply both $f(f^{-1}(x))$ and $f^{-1}(f(x))$; they should both return $x$.