The image shows a problem asking to find algebraic formulas for a function $f(x)$ and its inverse $f^{-1}(x)$ in terms of the input $x$. The function provided is:

$f(x) = 2x + 3$

Steps to find the inverse function $f^{-1}(x)$:

1. Start with the equation for $f(x)$: $y = 2x + 3$ Replace $f(x)$ with $y$ to simplify.

2. Swap $x$ and $y$ to begin solving for the inverse: $x = 2y + 3$

3. Solve for $y$ (this will give the inverse function): $x - 3 = 2y$ $y = \frac{x - 3}{2}$

Thus, the inverse function is:

$f^{-1}(x) = \frac{x - 3}{2}$

Final answer:

• $f(x) = 2x + 3$
• $f^{-1}(x) = \frac{x - 3}{2}$

Let me know if you'd like further details or have any questions!

Here are 5 related questions to explore:

1. How do we know that $f(x)$ and $f^{-1}(x)$ are truly inverses?
2. What is the geometric interpretation of a function and its inverse on a graph?
3. How can you verify that $f^{-1}(x)$ is the correct inverse of $f(x)$?
4. What properties must a function satisfy to have an inverse?
5. How does the domain and range of a function relate to its inverse?

Tip: You can check if two functions are inverses by composing them: $f(f^{-1}(x)) = f^{-1}(f(x)) = x$.