## Yes, the fact that x <= 2 instead of x <-2 will make a difference in the final answer for the inverse function.

To find the inverse function, let's first rewrite the given function with the correct inequality:

f(x) = -(x-2)^2, x <= 2

Now, let y = f(x):

y = -(x-2)^2

To find the inverse, we need to solve for x in terms of y.

1. Start by multiplying both sides by -1:

-y = (x-2)^2

2. Take the square root of both sides, being careful with the negative sign:

sqrt(-y) = x - 2

3. Finally, add 2 to both sides to isolate x:

x = sqrt(-y) + 2

Now, we have x in terms of y:

x(y) = sqrt(-y) + 2

Therefore, the inverse function of f(x) = -(x-2)^2, x <= 2 is:

g(x) = sqrt(-x) + 2

Now, let's check if g(x) is indeed the inverse function by confirming if g(f(x)) = x and f(g(x)) = x.

1. g(f(x)):

g(f(x)) = g(-(x-2)^2)

= sqrt(-(-(x-2)^2)) + 2

= sqrt((x-2)^2) + 2

= |x-2| + 2

Since the given inequality is x <= 2, we need to take the positive square root when calculating the inverse function. Therefore, we have:

g(f(x)) = x - 2 + 2

= x

2. f(g(x)):

f(g(x)) = -((sqrt(-x) + 2) - 2)^2

= -sqrt(-x)^2

= -(-x)

= x

Since we also have f(g(x)) = x, we can conclude that g(x) = sqrt(-x) + 2 is indeed the inverse function of f(x) = -(x-2)^2, x <= 2.

I hope this clarifies the solution. Let me know if you have any further questions!