Question

Finding Domain and Range Graphically A function f is given. (a) Use a graphing calculator to draw the graph of f. (b) Find the domain and range of f from the graph.

$f\left(x\right)=\sqrt{x-1}$

Short answer

(a) The graph of the function f is shown in Figure (1).

(b) The domain of the function is $\left[1,\infty \right]$and the range of the function is $\left[0,\infty \right]$.

Step-by-step solution

Part a. Step 1. Write down the given equation.

The given equation is

$f\left(x\right)=\sqrt{x-1}\mathrm{...}\left(1\right)$

Part a. Step 2. Procedure to plot a graph using graphing calculator.

The steps to plot the graph on graphic calculator is stated below.

• Press "plot function" screen. This screen is a list of equations, starting with “y =” that allows write the equation.
• Find the graphing window screen. This screen allows to determine the length of the x and y-axes.”
• Set up the size of the graphing window spaces.
• Call the graph function. Press the “graph” button to display the graph.

Part a. Step 3. Plot the graph.

The required graph is shown below.

Figure (1)

Thus, the graph of f is shown in Figure (1).

Part b. Step 1. Define domain and range.

Domain of a function$y=f\left(x\right)$ is the set of all x values for which the function is defined.

Range of a function$y=f\left(x\right)$ is the set of all yvalues for which the function is defined.

Part b. Step 2. Find the domain from the graph.

Consider the graph of$f\left(x\right)=\sqrt{x-1}$ shown in Figure (1).

From the graph it is observed that the function is defined for x in between$1$ to $\infty$.

So, the domain of the function is,

$\text{Domain}=\left[1,\infty \right]$.

Part b. Step 3. Find the range of the function.

Consider the graph of$f\left(x\right)=\sqrt{x-1}$shown in Figure (1).

From the graph it is observed that the function is defined for y in between$0$to $\infty$.

So, the range of the function is,

$\text{Range}=\left[0,\infty \right]$.

Thus, the domain of the function is $\left[1,\infty \right]$and the range of the function is width="52">0, ∞.