Domain range intervals; if you can

[name_ask17]

Homework Statement

Hello. Can someon help me understand the difference between these two problems?
#1: f(x)+3
#2: f(x+3)

The reason I want to know if becuase my question tells me to use the domain of f(x) as [4, 8] and the range as [2, 6]
Then it wants to know the domain and range of both of those problems.

My first question here is that I got [1, 5] as the domain, but I don't know if that pertains to the first or second problem. If it pertains to either of them, how do I do the other problem? Also, how do I find the range of both of these problems? Would it be [-1, 3] and again, for the first or second problem? I'm guessing both of my answers would go with the first, but yet again, I am not sure.[/quote]
How did you "get" [1, 5] as domain and [-1, 3] as range without know which problem you were working on? In #1, you must calculate f(x) and then add 3. You are told that you can only calculate f(x) for x in [4, 8] so the domain should be obvious. If f(x) gives values between 2 and 6 what values are f(x)+ 3?

If f(u) can be calculated for u in [4, 8] and u= x+ 3, what can x be? Which problem is that for? Well, the domain applies to the value inside parentheses in f() so which has x+3 inside the parentheses? Anything that is inside the parentheses- before you calculate f- affects the domain, anything outside affects the range.

If your knowledge allows you, can you asist me in this problem? I keep looking at it and just can't seem to figure it out. Please help.

Thanks in advance -Name_ask17

 

[vela]

You're given a function f which maps values in the domain [4, 8] to values in the range [2, 6]. That means any value you plug into f has to be between 4 and 8, and the value you get out of f will be somewhere between 2 and 6.

For example, if you had f(x²), what you'd know is that x² has to be between 4 and 8. If you have f(y+10), then y+10 has to be between 4 and 8, which means y has to be between -6 and -2. Whatever is between the parentheses has to be between 4 and 8.

Now look at the first function. Let's call it g(x). You have g(x) = f(x)+3. What values of x can you plug into g so that what gets plugged into f is between 4 and 8? Those values make up the domain of g.

Similarly, you have h(x) = f(x+3). What values of x can you plug into h so that what gets plugged into f is between 4 and 8?

 

[name_ask17]

what is the difference between f(x) + 3 and f(x+3)?

 

[Ivan92]

Let's assume f(x)= 2x. For f(x)+3, we plug in 2x for f(x) so it would be 2x+3. However, when it comes to f(x+3) you take what is inside the parenthesis and put it where the x is. So it would look like this f(x+3)=2(x+3). Hope this helps.