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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: \( f(f^{-1}(x)) = f^{-1}(f(x)) = x \).

Find algebraic formulas that express f and f^(-1) in terms of the input x.

Discover how to find the inverse of the linear function f(x) = 2x + 3. This tutorial provides a detailed step-by-step explanation suitable for high school students learning about inverse functions and algebraic manipulation.

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