Question

Domain and Range from a Graph A function f is given. (a) Sketch a graph of f. (b) Use the graph to find the domain and range of f.

$f\left(x\right)=3x-2$

Short answer

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

(b) The domain of the function is$\left(-\infty ,+\infty \right)$ and the range of the function is $\left(-\infty ,+\infty \right)$.

Step-by-step solution

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

The given equation is

$f\left(x\right)=3x-2\mathrm{...}\left(1\right)$.

Part a. Step 2. Find the coordinate at different points.

Substitute 0 for$x$ in equation (1).

role="math" localid="1646305914884" $$\begin{gathered} f\left(0\right)=3\left(0\right)-2 \\ =-2 \end{gathered}$$

So, the co-ordinate is $\left(0,-2\right)$.

Substitute 0 for $f\left(x\right)$in equation (1).

role="math" localid="1646305886000" $$\begin{gathered} 0=3x-2 \\ 3x=2 \\ x=\frac{2}{3} \end{gathered}$$

So, the co-ordinate is $\left(\frac{2}{3},0\right)$.

Part a. Step 3. Plot the graph.

Join the points to obtain the required 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$f\left(x\right)=3x-2$ shown in Figure (1).

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

So, the domain of the function is,

$\text{Domain}=\left(-\infty ,+\infty \right)$.

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

Consider the graph of$f\left(x\right)=2x+3$ shown in Figure (1).

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

So, the range of the function is,

$\text{Range}=\left(-\infty ,+\infty \right)$.