domain of sqrt(1-5^t)

Question

tkhunny · Answer

$$1 - 5^{t} \ge 0$$ You tell me why.

anonymous · Answer

huh

tkhunny · Answer

That was not a good answer.  You WILL have ro do better than that.

Find the Domain - Often, it is an exercise in figuring out what doesn't work.

Prep Question: What's the square root of -5?

anonymous · Answer

its nonreal

tkhunny · Answer

Excellent.  What values can we put in a square root and get REAL?

anonymous · Answer

positive numbers

tkhunny · Answer

Almost.  Let's include zero (0) while we're at it.

Now to the problem.  We've a square root with an argument.  That argument can't be negative.  Make sense?

anonymous · Answer

yes

tkhunny · Answer

Perfect.  Then $$1 - 5^{t} \ge 0$$  Still making sense?

anonymous · Answer

are you saying solve for t

tkhunny · Answer

Yes.  Do you have a plan?

anonymous · Answer

t is zero

tkhunny · Answer

t = 0 is part of the solution.  It's not an equality.  It an INequality.

anonymous · Answer

all real number

anonymous · Answer

except t lessthan or equal to 0

tkhunny · Answer

Now, I think you are guessing a little.  Let's use the mathematics.  If $$1 - 5^{t} \ge 0$$, then $$1\ge 5^{t}$$   Thus, we can pick any value of 't' such that $$5^{t}$$ does not exceed 1.  Still making sense?

anonymous · Answer

no

tkhunny · Answer

How did you decide on this?  "t lessthan or equal to 0"

anonymous · Answer

the restrictions in the radicand

tkhunny · Answer

Well, there you have it.

anonymous · Answer

I want to put it in interval notation

tkhunny · Answer

$$(\infty,0]$$  That's just a matter of practice.

anonymous · Answer

thanks

anonymous · Answer

how would you write x>ln5 in interval notation

tkhunny · Answer

$$(ln(5),\infty)$$

anonymous · Answer

how do you do that

tkhunny · Answer

Click the [Equation] button and play around with it.  Actually, I usually just write the code by hand if it's not too complicated.

anonymous · Answer

i mean how do you put it in interval notation

tkhunny · Answer

Well, the interval has endpoints, just list them.
Actually, I just lied.  The endpoints might be included.

3 < x < 4 is (3,4) 
3 < x <= 4 is (3,4]
3 <= x < 4 is [3,4)
3 <= x <= 4 is [3,4]

Infinity is NEVER "[" or "]".  You can't get there.  Always "(" or ")"

anonymous · Answer

ok but how did you get infinity for x>ln5

tkhunny · Answer

"Infinity" is a funny word.  It means too many things.  In this case, it means "there is no limit".  Since we've only x > ln(5), with what shall we fill in the other "end point"?

ln(5) < x < ??

anonymous · Answer

ok thanks