The function f(x)=(x+6)^2 is not one-to-one. Choose the largest possible domain containing the number 0 so that f(x) restricted to this domain is one-to-one.

The largest possible domain is [-6,infinity)
If the restriction function is written F, then inverse function F^{-1}(y)=...

The largest possible domain of [-6,∞) is correct for a one-to-one. In general, for a parabola, it is from the vertex to either +∞ or -∞.

Note that the domain of F becomes the range of F-1.

See previous post for hints on how to tackle this one.

http://www.jiskha.com/display.cgi?id=1288236842

F^{-1}(y) = -6 + \sqrt{y}

To find the inverse function of the restricted function F(x) = (x + 6)^2, we can follow these steps:

1. Replace F(x) with y: y = (x + 6)^2.
2. Swap the x and y variables: x = (y + 6)^2.
3. Solve the equation for y: Taking the square root of both sides, we get √x = y + 6.
4. Isolate y: Subtract 6 from both sides, giving y = √x - 6.

Therefore, the inverse function of F(x) when restricted to the largest possible domain [-6, infinity) is F^(-1)(y) = √y - 6.

To find the inverse function F^{-1}(y), we need to solve the equation for x in terms of y.

Given that f(x) = (x+6)^2, we set y = (x+6)^2 and solve for x.

Let's go through the steps to find F^{-1}(y):

Step 1: Set y = (x+6)^2.
y = (x+6)^2

Step 2: Take the square root of both sides to eliminate the exponent.
y^(1/2) = (x+6)

Step 3: Subtract 6 from both sides to isolate x.
y^(1/2) - 6 = x

Now, we have x in terms of y, and this expression represents the inverse function F^{-1}(y).

Therefore, the inverse function is F^{-1}(y) = y^(1/2) - 6.

Note: The domain of the inverse function is determined by the range of the original function. Since the original function f(x)=(x+6)^2 is not one-to-one, the range is [0, infinity). Therefore, the domain of the inverse function is [0, infinity).