Question

Find the domain and sketch the graph of the function. What is its range? $$ g(x)=\sqrt{x-1} $$

Short answer

The domain of the function \(g(x) = \sqrt{x-1}\) is \(x \geq 1\) or \([1,+\infty)\). After sketching the graph which starts at the point (1,0) and extends to the right and up, we find the range to be \(y\geq0\) or \([0,+\infty)\).

Step-by-step solution

Find the domain of the function

The function is \(g(x) = \sqrt{x-1}\). A square root of a quantity is a real number only when the quantity inside the square root is greater than or equal to zero. So, \(x-1 \geq 0\). Solve for \(x\): \[x \geq 1\] Therefore, the domain of the function is \(x\geq1\), or in interval notation, \([1,+\infty)\).

Sketch the graph of the function

To sketch the graph of the function \(g(x) = \sqrt{x - 1}\), you can begin plotting points on the basic square root graph \(y = \sqrt{x}\) and then observe the transformation that will occur when -1 is added within the square root function. Shift the points 1 unit to the right. This will give us: 1. (2,0) instead of (1,0) 2. (3,1) instead of (2,1) 3. (5,2) instead of (4,2) and so on. Now, connect these points in the typical square root shape. The graph will start at the point (1,0) and extend to the right and up, with x-values greater than or equal to 1.

Find the range of the function

Analyze the graph from step 2 to determine the range of the function. The graph starts at (1,0) and extends upwards. In other words, the function reaches every positive y-value, but it never goes below 0. Hence, the range of the function is \(y\geq0\) or in interval notation, \([0,+\infty)\).