Question

For the function y=ln(x-1)+2 which of the following statements is true.
A.) the domain is (-1,infinity) and the range is all real numbers
B.) The domain is (1, infinity) and the range is [2,infinity)
C.) The domain is real numbers and the range is [2,infinity)
D.) The domain is (1,infinity), and the range is all real numbers

Answer 1

well to find domain see u have ln (x-1)+2
when we have ln (g(x))for domain we have this condition g(x) >0
and range all vlue of in (g(x)) +2

Answer 2

so is it A since -1 is less than zero and you said the range is all real numbers?

Answer 3

Just a hint, you can't have negative numbers inside of logarithms.

ln(-1-1)=ln(-2)

The key is to find the number that makes it equal 0 inside of the logarithm.

Answer 4

so x=1

Answer 5

Correct, logarithms also can't have 0 in them either. So what you should do is to find what number makes the sum inside the logarithm = 0. Once you do that, you know the domain is any number above that number. In this case any number above 1 is the domain. However 1 is not part of the domain, it's all numbers above 1.

Answer 6

so you are saying that it is C?

Answer 7

No, the answer is either B or D.

Answer 8

Notice how for the domain it says (1, infinity) and for the range it says [2, infinity)? The key thing is to notice that there's a "(" next to the 1 and "[" next to the 2. This is because when you use "(" what you are saying is "All numbers above this number but NOT including this number" whereas when you use "[" you are saying "All numbers above this number including this number."

If you use ")" then you're saying "All numbers below this number but NOT including this number" and if you use "]" then you are saying "All numbers below this number including this number."

With infinities "(" and ")" are always used, you never use
"[" or "]" for infinities.

So the domain says (1, infinity) meaning that the domain is all numbers above one but NOT including 1, which is correct.

Answer 9

By the way I'm not implying the correct answer is B, i"m just trying to help you understand interval notion. Interval notation is the use of "(" and "]" to explain domains and ranges.

Answer 10

oh ok i really appreciate that

Answer 11

Yes so the correct answer is B or D.

Answer 12

so what about the 2 is that important?

Answer 13

The [2, infinity] is saying that the lowest y value ("range" simply means what are the possible y values of the function) is 2 and since 2 follows "[" that means that it is saying "All numbers above 2 including 2" and it's saying the highest y-value goes off into infinity.

Answer 14

oh ok so with the [ it implies that it is all real numbers after too so it could be B then? since that is where it starts and it isnt going the opposite of 2?

Answer 15

Do you have a calculator that can use ln?

Answer 16

yes i do i have graphing calculator ti inspire with me

Answer 17

Okay do this, plug in ln(1.01-1)+2 and see what you get, then do ln(1.001-1)+2 and see what you get, then ln(1.0001-1)+2 and see what you get. The point being that since the domain is all numbers above 1, we want to plug in numbers for "x" that get closer and closer to 1 but still are above 1.

When you do this in your calculator, you'll find that the y-values keep becoming smaller and smaller numbers (meaning you are getting bigger negative numbers as you approach 1 closer and closer).

Next step is to just plug in ln(2-1)+2 and you should get 2.

So what does this tell you?

Answer 18

it goes positive and in a straight line going both ways

Answer 19

I don't know what you mean by that. But what it should tell you is that the correct answer is D. We know the domain is all numbers above 1 which is (1, infinity). We did the test to show that as you plug in x-values closer and closer to 1 (1.01, 1.001, 1.0001, etc, etc) you get bigger and bigger negative numbers. When you plug in 2, you get 2 as your y-value. Meaning the range for the function is all real numbers.

Answer 20

Cool thank you for the help and i learned something today :D

Answer 21

You're welcome, have a nice day.