Question

Find the domain and the range of the function. $$f(x)=\sqrt{x}-3$$

Short answer

The domain of the function \(f(x) = \sqrt{x} - 3\) is \(x \geq 0\) and its range is \(y \geq -3\), or in interval notation: domain is [0, \(\infty\)) and range is [-3, \(\infty\)).

Step-by-step solution

Determine the domain

The domain of a function comprises the set of all possible input values (x values), which will generate real-number outputs. Since this function involves a square root, we are aware that the input value under the square root should always be greater than or equal to zero, otherwise we will end up with a complex number which is beyond the realm of real numbers. So, we establish the condition that \(x \geq 0\). Therefore, the domain of \(f(x) = \sqrt{x} - 3\) is \(x \geq 0\) or in interval notation [0, \(\infty\))

Determine the range

The range of a function is the set of all possible output values (y values). In this case, the smallest value under the square root is 0 (due to our domain restriction), and that gives us a minimal 'y' value of -3. Since the square root of 'x' can get arbitrarily large as 'x' increases, the function has no upper limit. Therefore the range of the function \(f(x) = \sqrt{x} - 3\) is all values greater than or equal to -3. Therefore, in interval notation the range is \([-3, \infty)\].