Will someone please point out what I am doing wrong?

Question

I have to find the domain and range in this problem:

y = radical (4 - x^2)

Domain:
4 - x^2 greater than or equal to 0
-x^2 greater than or equal to -4
x^2 less than or equal to 4
x less than or equal to +/- 2

My answer key says the answer is [-2, 2]. Why don't we use this same "+/-" logic for the range? LOOK BELOW.

Range:
y^2 = 4 - x^2
y^2 - 4 = -x^2
x^2 = -y^2 + 4
x = radical -y^2 + 4

-y^2 + 4 greater than or equal to 0
-y^2 less than or equal to -4
y^2 less than or equal to 4
y less than or equal to +/- 2/

I was marked wrong on a test and this was the right answer:

The *right* answer for this is simply y is less than or equal to 2 (no +/- sign). AKA (- infinity, 2]

WHY IS IT NOT LIKE THE DOMAIN?

Answer 1

In the first case you should write -2<=x <=2 and so on

In the second case -y^2 <=-4 implies y2>=4
so
y^2 should be >=2 or y^2<= -2 but the last can´t be because y^2>=0
and y^2 >=2 implies y>= sqrt2 and y<= -sqrt2
I hope that now you can go on

Answer 2

Will someone please point out what I am doing wrong?
I have to find the domain and range in this problem:

y = radical (4 - x^2)

Domain:
4 - x^2 greater than or equal to 0
-x^2 greater than or equal to -4
x^2 less than or equal to 4
x is less than or equal to +/- 2


Answer: Your last step is wrong. The correct conclusion is: modulus of x is less than or equal to 2 which means x is in
[-2,+2]


My answer key says the answer is [-2, 2]. Why don't we use this same "+/-" logic for the range? LOOK BELOW.

Range:
y^2 = 4 - x^2
y^2 - 4 = -x^2
x^2 = -y^2 + 4
x = radical -y^2 + 4

-y^2 + 4 greater than or equal to 0
-y^2 less than or equal to -4
y^2 less than or equal to 4
y less than or equal to +/- 2/

I was marked wrong on a test and this was the right answer:

The *right* answer for this is simply y is less than or equal to 2 (no +/- sign). AKA (- infinity, 2]

WHY IS IT NOT LIKE THE DOMAIN?


My solution to part (b):
y = radical (4 - x^2)
For x in [-2,+2] ( as proved above), the minimum value of y is 0 when x= +2 or -2 and its maximum value is 2 when x=0.
So the range is [0,2] which neither agrees with your answer or the so called 'right 'answer..

Answer 3

less than or equal to 2 includes -2

the range is different because you have to make sure the "x^2) gets no bigger than 4, or else you will have a negative number, which is impossible to take the square root of

Answer 4

i don't know.... but
from domain;
4 - x^2 >= 0
so x = 2...ans.....

so, y = 0, but the ans. also right if y <=2.... tq