# 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, ∞).