Question

Find the domain and the range of the function. Then sketch the graph of the function. $$y=\sqrt{x}-3$$

Short answer

The domain of the function is \([0, +\infty)\) and the range is \([-3, +\infty)\). The graph starts at the point (0,-3) and increases gradually as x increases but never drops below -3.

Step-by-step solution

Identify the Domain

Since we are dealing with a square root function, and the value under the square root, (x), must be greater than or equal to 0, the domain can be written as \(x \geq 0\). In interval notation, the domain is \([0, +\infty)\)

Determine the Range

The range of the function is determined by the possible output values. In this function, every output value will be the square root of x, subtracted by 3. As x>=0 and the smallest possible value for \( \sqrt{x} \) will be 0 (when x=0), the smallest possible value for y will be \( \sqrt{0} - 3 = -3 \). Therefore, the range of the function is Y: \([-3, +\infty)\)

Sketch the Graph

The graph of the function \(y = \sqrt{x} - 3\) is a horizontal shift of the graph of \(y = \sqrt{x}\) 3 units down. This means the graph starts at the point (0,-3), and as x moves away from 0, y will increase. This results in a graph that gradually increases but never drops below -3 as x increases.