ANSWERS: 2
  • The domain is the allowed values for x. From the question we could guess that x must be a real number between -4 and 4 inclusive, because a square root of a negative number would be imaginary, and if x is outside that range, that is what would happen. A function is a kind of map from one set to another. The definition of a function is not complete without specifying the sets in question, so asking for the domain without knowing these sets makes no sense. Are we allowed imaginary numbers or not? We need to know before hand. The range is the possible values for the function, which depends on the domain. If we assume the domain is the set of real numbers between -4 and 4 inclusive, then 16-x^2 is between 0 and 16 inclusive, and sqrt(16-x^2) is between 0 and 4 inclusive, which would be the range of f.
  • The answer depends on whether you want to restrict the answer to the set of all real numbers or you want to allow the use of complex numbers of the form: x = a + bi where i = √(-1), i^2 = -1 and the domains for a and b are -∞ < a < ∞, -∞ < b < ∞ Then the real number range for f(x) is: -∞ < f(x) < ∞ For example: If you choose the value x = 3i, then f(3i) = √(16-{3i}^2) f(3i) = √(16-{-9}) f(3i) = √(16+9) f(3i) = √(25) f(3i) = ±5 Also, you can choose numbers of the form x = a + 0i, then the range for f(x) is any imaginary number of the form: yi; -∞ < y < ∞ The only values that you cannot get for the range is a complex number of the form f(x) = a + bi where a <> 0 and b <> 0. Note: I am using <> for the not equal symbol.

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