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domain and function

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2 Answers

Hi Dalia;
Thank you for the update.  Let me try again.
h(x) = √(x) +6
 
 
The first issue to consider is that a figure which must be square-rooted, cannot be negative.  Henceforth, zero is the smallest number which can be square rooted.
 
√(x) must be equal or greater than zero.
 
Henceforth, 0 is the lowest number on the x-axis.
There is nothing limiting the x-axis in the positive direction.
The domain of x is...
0, infinity.
 
h(x)=y
This is the range, also known as the y-axis.
h(x) = √(x) +6
y=√(0) +6
y=6
 
The line begins on the y-axis, at its point of 6.
There is nothing limiting the y-axis in the positive direction.
Henceforth, the range is...
6, infinity
 
The first point of the line is
0, 6
There is no last point.
The line proceeds to the right, to infinity.
 
Please let me know if I can do anything more to help.
 
Dalia,
 
The Domain of a function are the "allowed" values that can be inserted into the function and obtain real numbers for answers, i.e. if the number 5 is in the domain, it means that you can put 5 into the function as a value for x.  Basically, there are only 2 kinds of values you cannot put into a function.  First, you cannot choose values for x that produce a zero in the denominator of a fraction, so for 1/x, the number 0 is not in the function's domain.  The second is that you cannot choose values for x that result in any negative numbers being square rooted.  This is the situation you have here.  If x were equal to -1, then you would have to take the square root of negative 1, which is imaginary, not real.  Any negative number is prohibited, so the smallest value for x is 0.  (FYI, if the 6 were inside the square root, then x could be as small as -6 before you would encounter values resulting in taking the square root of a negative number.  Texts and teachers often have preferred styles for reporting domains, but usually, they are represented using something like "All x, x ≥ 0"
 
The Range of a function are the outputs of the function, i.e. the values h(x) takes on as the values for its domain are inserted.  So what values of h(x) will you observe if you were to insert 0, .5, 1, 2, 3, .... to infinity?  The smallest number is not zero!  Can you get negative values for h(x)?  Again the range is expressed but now you use the label of the function, i.e. h(x) ≥ ___. 
 
I hope this helps.  Feel free to email me if it doesn't make sense or you have more specific questions.  John

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