Part 1a: Finding the Inverse Function

The function given is $f(x) = 2x + 7$. To find the inverse function, follow these steps:

1. Replace $f(x)$ with $y$: $y = 2x + 7$

2. Swap $x$ and $y$ (this is the key step to finding the inverse): $x = 2y + 7$

3. Solve for $y$: $x - 7 = 2y$ $y = \frac{x - 7}{2}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x - 7}{2}$

Part 1b: Verifying the Inverse Function

To verify that $f^{-1}(x) = \frac{x - 7}{2}$ is correct, we need to check two conditions:

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

   Substitute $f^{-1}(x)$ into $f(x)$: $f(f^{-1}(x)) = f\left( \frac{x - 7}{2} \right)$ Using the original function $f(x) = 2x + 7$: $f\left( \frac{x - 7}{2} \right) = 2\left( \frac{x - 7}{2} \right) + 7 = (x - 7) + 7 = x$ So, $f(f^{-1}(x)) = x$.

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

   Substitute $f(x) = 2x + 7$ into $f^{-1}(x)$: $f^{-1}(f(x)) = f^{-1}(2x + 7) = \frac{(2x + 7) - 7}{2} = \frac{2x}{2} = x$ So, $f^{-1}(f(x)) = x$.

Both conditions are satisfied, confirming that $f^{-1}(x) = \frac{x - 7}{2}$ is the correct inverse function.

Answer Choice:

The correct answer is: $\boxed{f^{-1}(x) = \frac{x - 7}{2}, \text{ for all } x.}$

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Would you like further clarification or details on any step?

5 Related Questions:

1. What are the general steps to finding an inverse of a linear function?
2. How can you verify if a function is one-to-one before finding its inverse?
3. Why is it necessary to swap $x$ and $y$ when finding an inverse function?
4. Can a function have an inverse if it is not one-to-one? Why or why not?
5. How would the process change if the function were not linear?

Tip:

To ensure a function is one-to-one, check that it passes the horizontal line test, meaning no horizontal line intersects the graph more than once.