state the domain of the following function in interval notation
f(x)= sqrt 2x-9/x-5
plug in numbers to explain reasoning
give step by step
The domain of the function can be determined by looking for any x-values that would make the denominator equal to zero, as division by zero is undefined.
Step by step:
1. Set the denominator equal to zero and solve for x:
x - 5 = 0
x = 5
2. Now, plug in numbers on either side of x = 5 to determine the domain:
- If x = 4 (x < 5):
f(4) = sqrt(2(4) - 9)/(4 - 5) = sqrt(8 - 9)/(-1) = sqrt(-1)/(-1)
Since the square root of a negative number is not real, f(4) is undefined.
- If x = 6 (x > 5):
f(6) = sqrt(2(6) - 9)/(6 - 5) = sqrt(12 - 9)/(1) = sqrt(3)/(1)
f(6) is defined and is equal to sqrt(3).
Therefore, the domain of the function f(x) is all real numbers except x = 5. In interval notation, the domain can be written as:
(-∞, 5) U (5, ∞)