Answer:
The range is negative numbers.
The interval for the range is
.
***You might want to look at your functions again because I don't see a choice that matches.
Step-by-step explanation:
Given functions:
![u(x)=-2x^2](https://tex.z-dn.net/?f=u%28x%29%3D-2x%5E2)
![v(x)=\frac{1}{x}](https://tex.z-dn.net/?f=v%28x%29%3D%5Cfrac%7B1%7D%7Bx%7D)
We are asked to find the range of
.
I'm also going to look at the domain just to see if this possibly might change my range .
is the inner function. So we will consider the domain of that function first.
You only have to worry about division by zero for the function
.
Since we are dividing by
, we don't want
to be zero.
So far the domain is all real numbers except
.
Now let's move out.
exists for all numbers,
. So we didn't want to include
from before.
Now let's put it together:
![(u \circ v)(x)](https://tex.z-dn.net/?f=%28u%20%5Ccirc%20v%29%28x%29)
![u(v(x))](https://tex.z-dn.net/?f=u%28v%28x%29%29)
![u(\frac{1}{x})](https://tex.z-dn.net/?f=u%28%5Cfrac%7B1%7D%7Bx%7D%29)
![-2(\frac{1}{x})^2](https://tex.z-dn.net/?f=-2%28%5Cfrac%7B1%7D%7Bx%7D%29%5E2)
![-2(\frac{1^2}{x^2})](https://tex.z-dn.net/?f=-2%28%5Cfrac%7B1%5E2%7D%7Bx%5E2%7D%29)
![-2(\frac{1}{x^2})](https://tex.z-dn.net/?f=-2%28%5Cfrac%7B1%7D%7Bx%5E2%7D%29)
![\frac{-2}{x^2}](https://tex.z-dn.net/?f=%5Cfrac%7B-2%7D%7Bx%5E2%7D)
So the domain is still all real numbers except at
since we cannot divide by 0 and
is 0 when
.
with
.
is positive for all numbers except
.
So
is negative for all numbers since negative divided by positive is negative.
So the range is only negative numbers.
Let's also look at the inverse:
![y=\frac{-2}{x^2}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B-2%7D%7Bx%5E2%7D)
Multiply both sides by
:
![yx^2=-2](https://tex.z-dn.net/?f=yx%5E2%3D-2)
Divide both sides by
:
![x^2=\frac{-2}{y}](https://tex.z-dn.net/?f=x%5E2%3D%5Cfrac%7B-2%7D%7By%7D)
Take the square root of both sides:
.
So
can't be 0 and it also can't be positive because the inside of the square root will be negative (since negative divided by positive results in negative).