Determine the domain and range of the function f(x):Ɍ→Ɍ such that
f(x)=(x^2)/√(x^2-4)
f(x)=(2x)/(x-2)(x+1)
For the function f(x) = (x^2)/√(x^2-4), the domain is all real numbers except for x values that make the denominator equal to zero. In this case, x cannot be equal to 2 or -2 since that would make the square root undefined. So the domain is all real numbers except -2 and 2.
For the range, we can consider the behavior of the function as x approaches positive infinity and negative infinity. As x approaches positive infinity, the function approaches 1, since the term (x^2)/(√(x^2-4)) simplifies to 1 when x is large. As x approaches negative infinity, the function approaches -1, since the term (x^2)/(√(x^2-4)) simplifies to -1 when x is large and negative.
Therefore, the range of the function is (-1, 1).
For the function f(x) = (2x)/((x-2)(x+1)), the domain is all real numbers except for x values that make the denominator equal to zero. In this case, x cannot be equal to 2 or -1 since that would make the denominator equal to zero. So the domain is all real numbers except -1 and 2.
To determine the range, we need to consider the behavior of the function as x approaches positive infinity and negative infinity. As x approaches positive infinity, the function approaches 0, since the numerator (2x) grows linearly while the denominator ((x-2)(x+1)) grows quadratically. As x approaches negative infinity, the function also approaches 0 for the same reasons.
Therefore, the range of the function is (-∞, ∞), meaning it can take any real value.
To determine the domain and range of the given functions, let's analyze each function separately:
1. f(x) = (x^2) / √(x^2 - 4):
Domain: The function is defined for all real numbers (Ɍ) except where the denominator is zero and the radicand under the square root sign is negative. So, to find the domain, we need to set the denominator and the radicand equal to zero and solve for x.
Denominator: x^2 - 4 ≠ 0
(x - 2)(x + 2) ≠ 0
This implies that x ≠ 2 and x ≠ -2. So, the function is defined for all real numbers except x = 2 and x = -2.
Radicand: x^2 - 4 ≥ 0
(x - 2)(x + 2) ≥ 0
The inequality is true for x ≤ -2 and x ≥ 2. However, as we already found that x ≠ 2, the inequality can be simplified to x ≤ -2.
Combining the restrictions, the domain of f(x) is (-∞, -2) U (-2, 2) U (2, ∞).
Range: To determine the range, we look at the function's behavior as x approaches positive and negative infinity. As x approaches infinity, the term x^2 dominates, and √(x^2 - 4) also approaches infinity. Similarly, as x approaches negative infinity, the term x^2 dominates, and the square root approaches infinity as well. Therefore, the range of f(x) is (-∞, ∞).
2. f(x) = (2x) / ((x - 2)(x + 1)):
Domain: The function is defined for all real numbers except where the denominator is zero. So, to find the domain, we need to set the denominator equal to zero and solve for x.
(x - 2)(x + 1) ≠ 0
This implies that x ≠ 2 and x ≠ -1. So, the function is defined for all real numbers except x = 2 and x = -1.
Range: To determine the range, we look at the behavior of the function as x approaches positive and negative infinity. As x approaches infinity, the term 2x dominates, and the denominator ((x - 2)(x + 1)) approaches infinity. Similarly, as x approaches negative infinity, the term 2x dominates, and the denominator approaches negative infinity. Therefore, the range of f(x) is (-∞, ∞).
In summary:
- For f(x) = (x^2) / √(x^2 - 4):
- Domain: (-∞, -2) U (-2, 2) U (2, ∞)
- Range: (-∞, ∞)
- For f(x) = (2x) / ((x - 2)(x + 1)):
- Domain: Ɍ - {2, -1} (all real numbers except 2 and -1)
- Range: (-∞, ∞)