SOLUTION: Suppose you have a function y = f(x) such that the domain of f(x) is 1 ≤ x ≤ 6 and the range of f(x) is −3 ≤ y ≤ 5. What is the domain of f(2(x ͨ
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Question 403290: Suppose you have a function y = f(x) such that the domain of f(x) is 1 ≤ x ≤ 6 and the range of f(x) is −3 ≤ y ≤ 5. What is the domain of f(2(x − 3))
Found 2 solutions by jim_thompson5910, robertb:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Since the domain is , this means that the set of possible input values is {1,2,3,4,5,6}. Say for the sake of argument that we're only dealing with integers (to make things easy for us)
Now consider that instead of plugging in 'x', you plug in x-3. So this means that instead of plugging in x=1, you'll plug x-3=1-3=-2 into the function.
So this means that the domain of f(x-3) is now
which becomes
So the domain of f(x-3) is
Now we're going to take it a step further. Instead of plugging in x-3, we're going to double it and plug in 2(x-3). So instead of plugging in -2, we're going to plug (-2)(2)=-4 into the function.
So this means that the domain becomes
So the domain of f(2(x-3)) is given that the domain of f(x) is .
Basically, the smallest number that the input 2(x-3) can be is -4 and the largest number that 2(x-3) can be is 6
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
The domain of f(2(x-3))can be determined in all generality, not only using discrete values.
Let g(x) = 2(x-3) = 2x - 6.
Then f(2(x-3)) = (f o g)(x) = f(g(x)). Let the domain of g be , its range . Similarly let the domain of f be , its range .
Then to find the domain of f o g we must find all x values in (which is the set of all real numbers), that are the pullback of intersection. Since the range of g is represented by 2(x-3), and the domain of f is the closed interval [1,6], we must then have
<==>
<==> .
Thus the domain of (f o g)(x) = f(2(x-3)) is the closed interval [7/2, 6].
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