Question

Find the domain and the range of the function. Then sketch the graph of the function. $$y=\sqrt{x-6}$$

Short answer

The domain of the function y = \( \sqrt{x - 6}\) is [6, +Infinity) and its range is [0, +Infinity). The graph is a curve starting at the point (6,0) and extending upwards and to the right.

Step-by-step solution

Find the Domain

The domain of a function f(x) = sqrt(x - a) is [a, +Infinity), as the expression under the square root cannot be negative. So, the domain for the given function \(y = \sqrt{x - 6}\) is [6, +Infinity). This implies that this function is defined for all real numbers greater than or equal to 6.

Find the Range

The range of a function f(x) = sqrt(x - a) is [0, +Infinity), because the square root always returns non-negative numbers. So, the range for the given function is [0, +Infinity). This means that the y-values that this function can take are all non-negative.

Sketch the Graph

To draw the graph, a table of values for the function can be created. The origin point of the graph will be at (6,0) since x=6 is the smallest number included in the domain. Then, the graph will extend upwards rightwards because as x increases, y also increases. The graph will be a curve that starts from (6,0) and goes upwards and to the right.