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Let's go through each part of the problem based on the function \( f(x) = 2x + 1 \).

### a) Find \( f(2) \)
We substitute \( x = 2 \) into the given function:

\[
f(2) = 2(2) + 1 = 4 + 1 = 5
\]

Thus, \( f(2) = 5 \).

### b) Find \( f^{-1}(x) \)
The inverse function \( f^{-1}(x) \) undoes the original function. To find the inverse, follow these steps:

1. Start with \( y = f(x) = 2x + 1 \).
2. Swap \( x \) and \( y \): \( x = 2y + 1 \).
3. Solve for \( y \):
   \[
   x - 1 = 2y
   \]
   \[
   y = \frac{x - 1}{2}
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x - 1}{2}
\]

### c) Find \( f^{-1}(7) \)
Now that we know \( f^{-1}(x) = \frac{x - 1}{2} \), substitute \( x = 7 \):

\[
f^{-1}(7) = \frac{7 - 1}{2} = \frac{6}{2} = 3
\]

Thus, \( f^{-1}(7) = 3 \).

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Would you like more details or clarifications on any part of this solution? Here are five related questions to expand on this topic:

1. What are the conditions for a function to have an inverse?
2. How can you verify that \( f^{-1}(x) \) is indeed the inverse of \( f(x) \)?
3. What is the domain and range of \( f(x) \) and \( f^{-1}(x) \)?
4. How do transformations (like shifts and stretches) affect the inverse function?
5. Can a quadratic function have an inverse? If so, under what conditions?

**Tip:** To check if two functions are inverses of each other, verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Given that f(x) = 2x + 1, a) Find f(2), b) Find f^{-1}(x), c) Find f^{-1}(7).

Learn how to solve a function problem involving finding f(2), the inverse of f(x), and evaluating f^{-1}(7). This step-by-step solution covers key algebraic concepts, including linear functions and inverse functions. Suitable for students in Grades 9-11.

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