find the domain of f(x)=5/¡Ì(x2-4)
and also range
To find the domain of the function f(x) = 5/√(x^2-4), we need to consider two conditions:
1. The expression inside the square root (x^2-4) should not be negative, as square root of a negative number is not defined in the real number system.
2. The expression should not result in division by zero, as division by zero is undefined.
For the first condition, x^2 - 4 >= 0, we solve the inequality:
x^2 - 4 >= 0
Factoring the expression:
(x-2)(x+2) >= 0
Now we consider the sign of each factor when the expression is greater than or equal to zero:
1. (x-2) >= 0: This is true when x >= 2.
2. (x+2) >= 0: This is true when x >= -2.
Combining the conditions, the domain of the function is x >= 2 or x <= -2.
To find the range of the function, we observe that as x approaches infinity or negative infinity, the value of f(x) approaches 0. Therefore, the range of the function is all real numbers except 0.
In summary:
Domain: x >= 2 or x <= -2
Range: All real numbers except 0.