How to Find Domain and Range of a Square Root Function

To find the domain and range of a square root function, we first need to understand the general form of the function, which is usually expressed as f(x) = √(x - k), where k is a constant.

Finding the Domain:

The domain of a square root function consists of all the values of x for which the function is defined. Since we cannot take the square root of a negative number in the realm of real numbers, we must set the expression inside the square root greater than or equal to zero:

x - k ≥ 0

This simplifies to:

x ≥ k

Thus, the domain of the function is all values of x that are greater than or equal to k. In interval notation, this is written as [k, ∞).

Finding the Range:

The range of a square root function is determined by the output values f(x). Since the square root function produces only non-negative outputs (i.e., it will always return 0 or a positive number), the smallest value of f(x) occurs when x = k:

f(k) = √(k - k) = √0 = 0

As x increases from k to infinity, f(x) also increases without bound. Thus, the range of the function is all values greater than or equal to 0. In interval notation, this is written as [0, ∞).

Example:

For the specific function f(x) = √(x - 4), the domain can be found by:

• Setting x - 4 ≥ 0, which gives x ≥ 4. Therefore, the domain is [4, ∞).

The range is determined as follows:

• The minimum value is f(4) = √(0) = 0, and since the function increases indefinitely, the range is [0, ∞).

In summary, to find the domain and range of a square root function, determine the values for which the expression under the root is non-negative for the domain, and recognize that the outputs will start from zero for the range.