Create an equation for the Area of the green area in terms of x . There are dire

Question

Create an equation for the Area of the green area in terms of x. There are directions for how to type math characters in the Resource Center that will show you show to enter your equation.

Use algebraic (symbolic) methods to determine the inverse of the function you found in (a). Explain how you got the inverse.

Use your graphing calculator to graph the inverse. Is the inverse a function? Explain your answer. If the inverse is not a function, what restrictions can be put on the graph to make it a function?

Explanation / Answer

Ok, so we know that the area of a square is just length times height.

So the length of the green area is (x-3), which is also the same as the height.

Now we have to take into account the area of the pool, which is just 4x4 = 16

So the total area will be the total area of everything minus the grey area and the pool in the middle.

So in terms of x we can write the following equation:

area = (length of everything minus grey length) x (width of everything minus grey width) - the pool area

which looks like this:
area = (x-3)(x-3) - 16

Multiply out and we get a quadratic equation that represents the area in terms of x:

Area = (x2 - 6x + 9) - 16   ===>   f(x) = x2 - 6x - 7

ok, so now we need the inverse of this function. Finding the inverse of a quadratic function is very tricky, but possible:

Rearrange the function so it takes the form y=a(x-h)2+k

So basically we have to complete the square to get it into that wierd form:

so take the value of the coefficient in front of x, half it, and square it, then add AND subtract it to the equation to get the following:

y = x2 -6x + [(3)2 - (3)2] -7

Because the first three terms are now a perfect square, you can write them in the form (a-b)2 or (a+b)2. The sign between the two terms will be the same as the sign of the coefficient of x in the equation

y = (x-3)2 -7

Here, a = 1; H = 3 (flip the sign of 3); k = -7

Now we solve for (x-3)2

(x-3)2 = y + 7

Now we solve for x by taking the square root of both sides and adding 3 to both sides as well:

x = (y+7) +3

and this is your inverse function

now if we graph this and use the vertical line test, we see that indeed it is a function. there is not a point were a vertical line goes through the graph of the function twice. This confirms that for any value of x, there is only one value of y. If it was not a function, we would restrict the domain such that it would pass the vertical line test.

Phew, that was long :)

Hope this helps!!

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