The function f be described by
The graph of this function, is the graph of the parent function
, which is the parabola with vertex at (0, 0) which opes upwards:
i) reflected with respect to the y-axis, because of the minus
ii) shifted 100 units up
check the picture attached.
A function has an inverse only if it is decreasing or increasing.
So we must divide the following cases for which the inverses exist:
i)
with domain (-infinity, 0]
ii)
with domain [0, infinity)
To find the formula for the inverse function g of f, we use the property:
f(g(x))=x
thus,
![f(g(x))=x\\\\ 100-[g(x)]^2=x\\\\g^2(x)=100-x\\\\g(x)= \mp\sqrt{100-x}](https://tex.z-dn.net/?f=f%28g%28x%29%29%3Dx%5C%5C%5C%5C%20100-%5Bg%28x%29%5D%5E2%3Dx%5C%5C%5C%5Cg%5E2%28x%29%3D100-x%5C%5C%5C%5Cg%28x%29%3D%20%5Cmp%5Csqrt%7B100-x%7D%20)
so each of - and + cases, are the inverses of
i)
with domain (-infinity, 0]
ii)
with domain [0, infinity)
At this point, recall that the domain of a function, is the range of it's inverse function,
and the range of the function, is the domain of the inverse function.
thus,
the range of
, which is [0,infinity) is the domain of
i)
with domain [0,infinity)
So our answer is:
inverse of i)
with domain (-infinity, 0] is
and the inverse of ii)
with domain [0, infinity) is
