The image you've uploaded appears to be focused on finding and verifying the inverse of a function $f(x)$. I'll guide you step-by-step through solving and filling in the missing information.

1. Interchanging $x$ and $y$:

The equation we start with is given as: $y = \frac{x + 2}{2}$ Here, the goal is to find the inverse of $f(x)$. To do this, we'll interchange $x$ and $y$, and then solve for $y$.

2. Replace $y$ with $f^{-1}(x)$:

After interchanging $x$ and $y$, we get: $x = \frac{y + 2}{2}$ Now, solve for $y$ (which will give us $f^{-1}(x)$): $2x = y + 2 \quad \text{(Multiplying both sides by 2)}$ $y = 2x - 2$ Therefore, the inverse function is: $f^{-1}(x) = 2x - 2$

3. Verify that $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$:

Let's first check $f^{-1}(f(x))$ and then $f(f^{-1}(x))$.

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

We know that: $f(x) = \frac{x + 2}{2}$ So, apply $f^{-1}(x)$ to $f(x)$: $f^{-1}(f(x)) = f^{-1}\left( \frac{x + 2}{2} \right)$ Substitute the formula for $f^{-1}(x)$: $f^{-1}\left( \frac{x + 2}{2} \right) = 2 \cdot \frac{x + 2}{2} - 2$ Simplify: $2 \cdot \frac{x + 2}{2} = x + 2$ $(x + 2) - 2 = x$ Thus, $f^{-1}(f(x)) = x$.

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

We know that $f^{-1}(x) = 2x - 2$, so we substitute this into $f(x)$: $f(f^{-1}(x)) = f(2x - 2)$ Apply the formula for $f(x)$: $f(2x - 2) = \frac{(2x - 2) + 2}{2}$ Simplify: $\frac{(2x - 2) + 2}{2} = \frac{2x}{2} = x$ Thus, $f(f^{-1}(x)) = x$.

Both verifications are successful, confirming that $f^{-1}(x) = 2x - 2$ is correct.

Would you like more details on any specific step, or have any questions?

5 Related Questions:

1. How do you find the inverse of more complicated functions?
2. What are the conditions for a function to have an inverse?
3. Can a function have more than one inverse?
4. How do you graph a function and its inverse on the same coordinate plane?
5. What are some real-world applications of inverse functions?

Tip:

Always ensure the function you are working with is one-to-one (i.e., passes the horizontal line test) before attempting to find an inverse.