Determine the domain and rage of the following functions F:R→R such that

Question

1/√(9-x^2)
3x/(x-2)(x+1)
√((x)/(2-x))
X^2-1/√4x^3

Answer 1

The domain and range of each function are as follows:

1) f(x) = 1/√(9-x^2):

Domain: The function is defined for all real numbers except when the expression under the square root becomes negative. Therefore, the domain of f(x) is (-∞, -3] ∪ [3, ∞).

Range: Since the square root of a non-negative number is always non-negative, the range of f(x) is [0, ∞).

2) f(x) = (3x)/((x-2)(x+1)):

Domain: The function is defined for all real numbers except when the denominator equals zero. Therefore, the domain of f(x) is (-∞, -1) ∪ (-1, 2) ∪ (2, ∞).

Range: The function is a rational function and can take any real value except when it approaches a vertical asymptote. By analyzing the behavior of the function as x approaches the asymptotes, we can determine that the range of f(x) is (-∞, -∞) ∪ (-∞, ∞) ∪ (-∞, ∞).

3) f(x) = √(x/(2-x)):

Domain: The function is defined for all real numbers except when the expression under the square root becomes negative and when the denominator becomes zero. Therefore, the domain of f(x) is (-∞, 0) ∪ (0, 2).

Range: The square root of a non-negative number is always non-negative. Therefore, the range of f(x) is [0, ∞).

4) f(x) = (x^2-1)/√(4x^3):

Domain: The function is defined for all real numbers except when the expression under the square root becomes negative and when the denominator becomes zero. Therefore, the domain of f(x) is (-∞, 0) ∪ (0, ∞).

Range: The range of f(x) depends on the behavior of the expression as x approaches positive and negative infinity. If we approach positive infinity, the function tends to zero. If we approach negative infinity, the function approaches negative infinity. Therefore, the range of f(x) is (-∞, 0].