Question

Which equation is the inverse of y = 100 – x2?

Answer 1

Answer:

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,

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

Step-by-step explanation:

Answer 2

The function f be described by $y=100- x^{2}$

The graph of this function, is the graph of the parent function $x^{2}$, 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) $y=100- x^{2}$ with domain (-infinity, 0]

ii) $y=100- x^{2}$ with domain [0, infinity)


To find the formula for the inverse function g of f, we use the property:

f(g(x))=x

thus, 

$f(g(x))=x\\\\ 100-[g(x)]^2=x\\\\g^2(x)=100-x\\\\g(x)= \mp\sqrt{100-x}$



so each of - and + cases, are the inverses of 


i) $y=100- x^{2}$ with domain (-infinity, 0]

ii) $y=100- x^{2}$ 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 $\sqrt{100-x}$, which is [0,infinity) is the domain of 

i) $y=100- x^{2}$ with domain [0,infinity)



So our answer is:

inverse of i) $y=100- x^{2}$ with domain (-infinity, 0] is $-\sqrt{100-x}$


and the inverse of ii) $y=100- x^{2}$ with domain [0, infinity) is 

$\sqrt{100-x}$