f(x)= square root x-4
I have to find the domain
f(x) = square root x-1/x-3
Thanks!
To find the domain of a function, we need to determine the values of x for which the function is defined.
For f(x) = √(x - 4), the square root is defined only for non-negative values. Therefore, the expression x - 4 must be greater than or equal to zero.
x - 4 ≥ 0
Solving this inequality, we add 4 to both sides:
x ≥ 4
Therefore, the domain of f(x) = √(x - 4) is all real numbers greater than or equal to 4, or in interval notation, [4, ∞).
For f(x) = √((x - 1)/(x - 3)), we need to consider two conditions:
1. The denominator, x - 3, cannot be equal to zero since division by zero is undefined. Therefore, x - 3 ≠ 0, which implies x ≠ 3.
2. The expression (x - 1)/(x - 3) within the square root cannot be negative, as the square root is defined only for non-negative values.
(x - 1)/(x - 3) ≥ 0
To solve this inequality, we consider the sign changes in the expression (x - 1) and (x - 3):
- For x < 1, both (x - 1) and (x - 3) are negative, so their division is positive.
- For 1 < x < 3, (x - 1) is positive, and (x - 3) is negative, so their division is negative.
- For x > 3, both (x - 1) and (x - 3) are positive, so their division is positive.
Therefore, the inequality (x - 1)/(x - 3) ≥ 0 holds for x < 1 and x > 3.
Combining both conditions, we find that the domain of f(x) = √((x - 1)/(x - 3)) is (-∞, 1) ∪ (3, ∞).