Domain
The domain of a function is the set of inputs for which the function gives a real output.
a)
We can see that the absolute value function gives a real output for all real inputs. Therefore, its domain is:
x ∈ ℝ or (-∞, ∞)
b)
We can see that the square root of (4 - y²) function gives a real output for all y-values from -2 to 2 (because √4 = 2). Therefore its domain is:
-2 ≤ y ≤ 2 or [-2, 2]
Range
The range of a function is its set of real outputs.
a)
We can see that the absolute value function provides outputs from y = -1 to infinity. Therefore, its range is:
y ≥ -1 or [-1, ∞)
b)
We can see that the square root of (4 - y²) function provides outputs from x = 0 to x = 2. Therefore, its range is:
0 ≤ x ≤ 2 or [0, 2]
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Note
The function in problem a is a function of x, meaning its input is x, whereas the function in problem b is a function of y, meaning its input is y.
This is because a function can have only one output for every input (think about the vertical and horizontal line tests).