state the domain of the following function in interval notation
g(x)= 3^sqrt 2x-5/x^2-4
explain how this is the answer
give step by step
The domain of a function is the set of all possible inputs for which the function is defined. In the given function, we have a square root in the numerator and a denominator that cannot be equal to zero.
To find the domain of the function g(x)= 3^(sqrt(2x-5))/(x^2-4), we need to consider the restrictions on the square root term and the denominator.
1. The expression under the square root, 2x-5, must be non-negative:
2x - 5 >= 0
2x >= 5
x >= 5/2
This sets a lower limit on the possible values of x.
2. The denominator, x^2-4, cannot be equal to zero:
x^2 - 4 ≠ 0
x^2 ≠ 4
x ≠ ±2
This sets additional restrictions on the values of x.
Combining both conditions, we find that the domain of the function is all real numbers greater than or equal to 5/2 and not equal to ±2. In interval notation, we can express this as:
Domain: (-∞, -2) ∪ (-2, 2) ∪ (2, 5/2] ∪ (5/2, ∞)