Question

Find the domain and the range of the function. Then sketch the graph of the function. $$y=7 \sqrt{x}$$

Short answer

The domain of the function y=7√x is [0,∞) and the range is also [0,∞). The graph is a vertically stretched version of the basic square root function that starts at the origin and extends indefinitely to the right as x increases.

Step-by-step solution

Identify the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since the square root function is undefined for negative values, the domain of y=7√x is x≥0, or, in interval notation, [0,∞).

Identify the Range

The range of a function is the set of all possible output values (y-values) that we get after applying the function to the domain. For y=7√x, since the square root of any non-negative real number is non-negative and we multiply it by 7, the range is y≥0, or in interval notation, [0,∞).

Sketch the Graph

The graph of y=√x starts at the origin (0,0) and extends indefinitely to the right, growing slowly over time. Here, we have y=7√x meaning that each y-value will be seven times the corresponding y-value for the basic square root function. This stretches the graph vertically by a factor of 7. So, the graph will start from the origin (0,0), rise more steeply than the basic square root graph, and extend indefinitely to the right.