Question

Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$ h(x)=\sqrt{x-1} $$

Short answer

The domain of the function \(h(x)=\sqrt{x-1}\) is \(x ≥ 1\), and its range is \(y ≥ 0\). The graph begins from the point (1,0) and increases as x increases, verifying that it aligns with a square root function shifted one unit to the right.

Step-by-step solution

Determine the Domain

Since the square root function exists for values which are zero or positive, it is inferred that the expression under the square root, i.e., \(x - 1\), should be greater than or equal to 0. Hence, we solve the inequality \(x - 1 ≥ 0\) which gives \( x ≥ 1\). So, the domain of the function \(h(x)\) is \(x ≥ 1\). This means the function exists for all x-values greater than or equal to 1.

Determine the Range

As the square root function results in positive values including zero, the range of \(h(x)\) is \( y ≥ 0 \). So, the function will yield all y-values greater than or equal to zero.

Sketch the Graph

Using the domain (x ≥ 1) and range (y ≥ 0), plot a few points on the graph to get a feel for the shape of the function. For instance, at x=1, h(x)=0; at x=2, h(x)=1; and at x=3, h(x) is approximately 1.41 (or \( \sqrt{2} \)). These points suggest that the function begins from the point (1,0) and increases slowly. Note that this is only a sketch and it is recommended to use a graphing utility to obtain a more precise graph.