Definition of function domain and range

  • Thread starter Zalajbeg
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I am a bit confused about this matter.

While I was studying Calculus I saw an excercise like this:

The domain of f(x) [0,2] and the range is [0,1], it also shows its graphic, though it is not important it is something like a parabola, its maximum point is (1,1) and its intersection points are (0,0) and (0,2).

It asks me to show domains and ranges of some other functions. And one of them is that:

f(x+1)

It is just this, it doesn't write it like y=f(x+1) but I assume it.

Then the answer is that in solution appendix. Domain: [-1,1] and Range: [0,1]

I would be ok with it if it wrote Domain: x=[-1,1]. However if it asks the domain of the function, isn't it still the same function "f"? Shouldn't it have the same domain for its argument?

It may sound like a very small detail or useless but I really wonder your opinions.
 

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  • #2
eumyang
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f(x) and f(x+1) are usually NOT the same function. Are you familiar with function transformations? The graph of f(x+1) is the graph of f(x) with a horizontal shift one unit to the left. So if the domain of f(x) is [0, 2], a horizontal shift of one unit to the left would mean that the domain of f(x+1) is [-1, 1]. The range would not change.

A couple more examples:
f(x-4) would be a horizontal shift of f(x) 4 units to the right, so the domain would be [4, 6]. The range would still be [0, 1].

f(x) + 3 would be a vertical shift of f(x) 3 units up. This time, the domain wouldn't change ([0, 2]), but the range would change to [3, 4].
 
  • #3
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f(x) and f(x+1) are usually NOT the same function. Are you familiar with function transformations? The graph of f(x+1) is the graph of f(x) with a horizontal shift one unit to the left. So if the domain of f(x) is [0, 2], a horizontal shift of one unit to the left would mean that the domain of f(x+1) is [-1, 1]. The range would not change.

A couple more examples:
f(x-4) would be a horizontal shift of f(x) 4 units to the right, so the domain would be [4, 6]. The range would still be [0, 1].

f(x) + 3 would be a vertical shift of f(x) 3 units up. This time, the domain wouldn't change ([0, 2]), but the range would change to [3, 4].
Thanks for your kind answer.

However I know all the things above. I am just confused with the notation.

I have read in Calculus and Analytical Geometry 9th edition from Thomas and Finney that:

"f(x)" means the value of the function at the argument of the function is equal to x, "f" means the function.

Therefore I believe f(x) and f(x+1) are not equal but both of them are elements of the same function f.

Then I think (regarding to my example above) if it asks the domain of the function I can understand it asks me the values which can be argument of it. Therefore as both of them are the function "f", the domain must be the same.

Doesn't domain of a function mean the set of elements which can be argument of the function?

But if it is said g(x)=f(x+1), and what the domain of the "g". It is ok domain is [-1,1]

May be they wanted to state that, it is a new function like this and what is its domain by saying show the domain of the function below.
 
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