Step-by-step explanation:
Si first let graph this function.
Start at endpoint (x=0),
[tex]f(0) = - (0) {}^{2} + 6 = 6[/tex]
Next, let's use two points
[tex]f(1) = - ( {1}^{2} ) + 6 = 5[/tex]
[tex]f(2) = - (2) {}^{2} + 6 = 2[/tex]
[tex]f(3) = - (3) {}^{2} + 6 = - 3[/tex]
[tex]f(4) = - (4) {}^{2} + 6 = - 10[/tex]
So using these 4 points
(0,6)
(1,5)
(2,2)
(3,-3)
(4,-10)
So plot these points and draw a parabolic curve starting at x=0 going infinity in the positive x direction.
As you can see, this function is one to one thus it has an inverse which is simply found by swapping the x coordinates and y coordinates.
Algebraically,
[tex]f(x) = - {x}^{2} + 6[/tex]
[tex]x = - {y}^{2} + 6[/tex]
[tex]x - 6 = - {y}^{2} [/tex]
[tex]6 - x = {y}^{2} [/tex]
[tex] \sqrt{6 - x} = f {}^{ - 1} (x)[/tex]
The range of f is y<=6, thus the domain of the inverse is x<=6.
The inverse function points would just be
(6,0)
(5,1)
(2,2)
(-3,3)
(-10,4)
The graph of both functions are shown below
If we scribble the line y=x between these two functions, they show a clear reflection about y=x.
