What is the domain and range of the square root of (x-3) 5
To find the domain and range of the function f(x) = √((x-3)*5), you need to consider the restrictions on x and the possible values of y.
The domain refers to all the possible values that x can take. In this case, since you are taking the square root of (x-3), the expression inside the square root must be non-negative. So you need to find the values of x that make (x-3)*5 ≥ 0.
To solve this inequality, you can use the fact that a product is positive when both factors have the same sign (either both positive or both negative). When x-3 > 0, or x > 3, the expression (x-3) is positive, and multiplying by 5 doesn't change the sign. Similarly, when x-3 < 0, or x < 3, the expression (x-3) is negative, and multiplying by 5 changes the sign.
Therefore, the domain of f(x) is x ∈ (-∞, 3] ∪ [3, +∞). This notation means that x can be any value less than or equal to 3, or any value greater than or equal to 3.
The range refers to all possible values that f(x) can take. Since you are taking the square root of (x-3)*5, the value of f(x) will always be greater than or equal to 0. So the range of f(x) is y ∈ [0, +∞). This notation means that y can be any value greater than or equal to 0.
In summary, the domain of f(x) is x ∈ (-∞, 3] ∪ [3, +∞) and the range of f(x) is y ∈ [0, +∞).