Answer:
a. domain: (-∞, -2] ∪ [2, ∞); range: [0, ∞)
b. domain: (-∞, ∞); range: [2, ∞)
Step-by-step explanation:
In each case, the domain is the set of x-values for which the function is defined. A square root function will be defined where its argument is non-negative.
The range of the function is the set of values it can produce as output. A square root function cannot produce negative values. The minimum value it can produce will depend on the argument.
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a. The function is defined where ...
x² -4 ≥ 0
x² ≥ 4
|x| ≥ 2 . . . . take the square root
x ≤ -2 ∪ 2 ≤ x . . . . . the domain of the function
The value of x² -4 can be any non-negative number, so ...
0 ≤ y < ∞ . . . . . the range of the function
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b. The function is defined where ...
x² +4 ≥ 0
True for all values of x.
-∞ < x < ∞ . . . . . the domain of the function
The value of x² +4 cannot be less than 4, so the function value cannot be less than √4 = 2.
2 ≤ x < ∞ . . . . . the range of the function
