Math  /  Algebra

QuestionIdentify the function that has a domain of x2x \leq-2 and a range of y3y \geq 3. a) y=x+2+3y=\sqrt{x+2}+3 b) y=(x+2)+3y=\sqrt{-(x+2)}+3 c) y=x23y=-\sqrt{x-2}-3 d) y=(x2)3y=-\sqrt{-(x-2)}-3

Studdy Solution

STEP 1

What is this asking? Which of these square root functions has the given allowed xx values (domain) and allowed yy values (range)? Watch out! Remember that the square root of a negative number isn't a real number, so we need to make sure what's *inside* the square root is zero or positive!

STEP 2

1. Analyze option (a)
2. Analyze option (b)
3. Analyze option (c)
4. Analyze option (d)

STEP 3

Alright, let's **start** with option (a): y=x+2+3y = \sqrt{x+2} + 3.
We want to see if its domain is x2x \leq -2.

STEP 4

If we plug in x=2x = -2, we get 2+2+3=0+3=3\sqrt{-2+2} + 3 = \sqrt{0} + 3 = 3.
That works!

STEP 5

If we plug in x=1x = -1 (which is *greater* than 2-2), we get 1+2+3=1+3=4\sqrt{-1+2} + 3 = \sqrt{1} + 3 = 4.
That also works!

STEP 6

But the problem says the domain should be x2x \leq -2.
So, x=1x = -1 shouldn't work.
Option (a) is **not correct**!

STEP 7

Now, let's **check out** option (b): y=(x+2)+3y = \sqrt{-(x+2)} + 3.

STEP 8

If we plug in x=2x = -2, we get (2+2)+3=0+3=3\sqrt{-(-2+2)} + 3 = \sqrt{0} + 3 = 3.
Looks good so far!

STEP 9

What about x=3x = -3?
We get (3+2)+3=(1)+3=1+3=4\sqrt{-(-3+2)} + 3 = \sqrt{-(-1)} + 3 = \sqrt{1} + 3 = 4.
Since 3-3 is less than 2-2, this is consistent with the domain x2x \leq -2.

STEP 10

Let's see about the range.
The smallest value that (x+2)\sqrt{-(x+2)} can be is **zero**, which happens when x=2x = -2.
So, the smallest value of yy is 0+3=30 + 3 = 3.
Since the square root part can't be negative, the function will always be *greater than or equal to* 33, so the range is y3y \geq 3.
This **matches** what we want!

STEP 11

Time for option (c): y=x23y = -\sqrt{x-2} - 3.

STEP 12

If we try x=2x = -2, we get y=223=43y = -\sqrt{-2-2} - 3 = -\sqrt{-4} - 3.
Uh oh!
We can't take the square root of a negative number, so this isn't going to work.
Option (c) is **out**!

STEP 13

Finally, let's **look at** option (d): y=(x2)3y = -\sqrt{-(x-2)} - 3.

STEP 14

If we try x=2x = -2, we get y=(22)3=43=23=5y = -\sqrt{-(-2-2)} - 3 = -\sqrt{4} - 3 = -2 - 3 = -5.

STEP 15

The problem says the range should be y3y \geq 3, but we got y=5y = -5.
So, option (d) is **incorrect**!

STEP 16

Option (b), y=(x+2)+3y = \sqrt{-(x+2)} + 3, is the **correct** answer!
It has the domain x2x \leq -2 and the range y3y \geq 3.

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