Domain: [7, ∞)
Range: [9, ∞)
Step-by-step explanation:
1) This question may be easily answered if you are aware of the shape of the graph of sq. root x, and the effect of translations to graphs.
1. x - 7 means that the original graph is translated 7 units in the positive direction of the x-axis (ie. to the right), thus the minimum value for x is also shifted from 0 to 7
2. the + 9 means that the graph is translated 9 units in the positive direction of the y-axis (ie. up), thus the minimum y-value is also shifted from 0 to 9
3. We know that the graph will continue to infinity, both in the x- and y-direction
Thus, the domain would be [7, ∞) and the range [9, ∞)
2) Another way to think about it is to ask yourself when it would make sense for the graph to exist. For this, we must consider that you cannot take the square root of a negative number.
Thus, if we have the square root of (x - 7), for what value of x would (x - 7) be negative? If x = 7:
x - 7 = 7 - 7 = 0
Therefor, any x-value less than 7 will lead to a negative answer, which wouldn't be practical. Any value equal to or greater than 7 will lead to a positive answer, thus the permissible values for x are from 7 to infinity, and so the domain is [7, ∞) (note that square brackets are used for 7 as it is included in the domain, whereas infinity is always closed with round brackets).
If we have already found the domain, then we can simply substitute the values for this into the equation to obtain the range (note that this will work for a square root function, however some functions will have turning points and in this case you must calculate the range based on the turning points as well as the minimum and maximum x-values):
if x = 7: y = sq. root (7-7) + 9
= 9
This is the minimum value for y
if we have x = ∞, then the y value will also be infinitely great, therefor the maximum y-value is also ∞. Thus, the range is [9, ∞).
These questions are much easier to solve however if you are already aware of the basic graph and the effect of dilations, translations and reflections so that you may visualise it better.
