Answer:
To summarize:
√3 - 6x: Domain is all real numbers (-∞, +∞).
√13 - (13 - 2x): Domain is all real numbers (-∞, +∞).
1/√x - 2: Domain is all real numbers except x = 0, represented as (-∞, 0) U (0, +∞).
Explanation:
To find the domain of each expression, we need to identify any restrictions or limitations on the variables that would result in undefined values.
√3 - 6x:
Since square roots are defined for non-negative numbers, the expression is defined as long as the value inside the square root (√3) is non-negative. The square root of 3 is a positive value, so there are no restrictions on the domain of this expression. Therefore, the domain is all real numbers (-∞, +∞).
√13 - (13 - 2x):
Similar to the previous expression, the square root (√13) is defined for non-negative values. However, we also need to consider the expression (13 - 2x) inside the parentheses. This expression does not have any limitations since it is defined for all real numbers. Therefore, the domain of this expression is all real numbers (-∞, +∞).
1/√x - 2:
For this expression, we need to consider both the square root (√x) and the denominator (1/√x). The square root (√x) is defined for non-negative values of x. However, the denominator 1/√x will become undefined if the value of x is zero because division by zero is undefined. Therefore, we need to exclude x = 0 from the domain. For all other non-zero values of x, the expression is defined. Therefore, the domain of this expression is all real numbers except x = 0, represented as (-∞, 0) U (0, +∞).