Question

HELP MEDAL

Answer 1

Find the domain of f(x) =

Answer 2

@phi @ganeshie8

Answer 3

and also Determine the range of

Answer 4

the domain are the "allowed" x values.

usually x is allowed to be *all numbers* i.e. all real numbers, written
\( -\infty < x < \infty\)

but sometimes we have to not allow some numbers because they cause trouble. The trouble could be:
1) divide by zero (not allowed)
2) square root of a negative number (not allowed if we want the range to be real)
3) log of negative number (ditto as in 2)

Answer 5

so for this, x is greater than -2 right?

Answer 6

for the domain question

Answer 7

in your problem, of the 3 items in my list, which 2 do not apply?
You can see we are not dividing, so it's not possible to divide by zero
Also, we are not taking the logarithm of anything.

That leaves only item 2 as a possible problem.

Answer 8

You have to figure out if it's possible to
take the square root of a negative number

depending on what x is in your problem

any ideas?

Answer 9

correct

Answer 10

no you can not take the square root of a negative number

Answer 11

oh so x has to be greater than 2

Answer 12

these are my options

Answer 13

the question is, can you find a value of x that makes the number inside the square root negative?

if you had *just*
\[ \sqrt{x+2}\]
and you put in x= -3
\[ \sqrt{-3+2} = \sqrt{-1} \]
and that is trouble

Answer 14

but you have
\[ \sqrt{(x+2)^2} \]

Answer 15

so it would be A (:

Answer 16

because it wouldnt be negative right?

Answer 17

if you replace x with 0, what do you get?

Answer 18

4

Answer 19

you get \[ \sqrt{4} \]
and that is ok.
so 0 is allowed.

if you say x has to be bigger than 2 (choice A), you are ignoring some of the domain.

Answer 20

what do you get if x= -3 ?

Answer 21

sqrt 1

Answer 22

because (-3+2)^2 = (-1)^2 = -1 * -1 = 1
and sqrt(1) is ok

in other words, you found an x < -2 which works.
that means the first 3 choices can't be correct (those choices do not allow the domain to include -3, and we found x=-3 is allowed)

Answer 23

so all real numbers
?

Answer 24

If you think about it, because you square (x+2), that number will always be positive. and that means we will never get in trouble, no matter what number we use for x

Answer 25

yay(: so its all real numbers

Answer 26

and can you just tell me if this is right?

Answer 27

the range are all the numbers that f(x) can be.

the (x+1)^2 part will always be zero or positive. It can never be negative (meaning never less than 0)
so the smallest f(x) will be is 0 -2 = -2
and f(x) can be bigger than -2
so
f(x)≥ -2
which means the range is y≥-2

Answer 28

f(x)>= -2
y>= -2

Answer 29

yay i ws right thanks (: