Q. 30 - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=\left\{\begin{array}{l}3x if x\ne 0\\ 4 if x=0\end{array}\right.$

(a) Find the domain of each function

(b) Locate any intercepts

(c) Graph each function

(d) Based on the graph,Find the range

(e) Is f continuous on its domain?

Step-by-Step Solution

Answer

(a) The domain of the function is the set of all real numbers as there is no value of  where the function is not defined. 

(b) The point $(0,4)$ lies on the graph.

(c)  The graph of the function

(d) The range is $\left\{y|y\ne 0\right\} or (-\infty ,0)\cup (0,\infty )$

(e) The function is discontinous at $x=0$

Step 1.Given information

The given function $f\left(x\right)=\left\{\begin{array}{l}3x if x\ne 0\\ 4 if x=0\end{array}\right.$

Step 2.Find the domain of each function

We will use the defination of given function

$f\left(x\right)=\left\{\begin{array}{l}3x if x\ne 0\\ 4 if x=0\end{array}\right.$

From the above definition we see the domain of the function is the set of all real numbers as there is no value of  where the function is not defined. 

Step 3.Locate any intercepts

The intercepts are the points on the graph which are obtained when it cuts the x and y axes.
The point $(0,4)$ lies on the graph. Hence the function has a y-intercept at $(0,4)$.
Also, the value of becomes 0 only when the value of x is 0 . This means that the x- intercepts is of length 0 . 

Step 4.Graph each function

The graph of the function

Step 5.Based on the graph,Find the range

From the graph we see the point $(0,0)$ do not lie on the graph. So the range of the function $f\left(x\right)$ is the set of real numbers except $y=0$ . Hence the range is  $\left\{y|y\ne 0\right\} or (-\infty ,0)\cup (0,\infty )$
.

Step 6.Checking f continuous on its domain?

From the graph we see the point $(0,0)$ do not lie on the graph. Hence the function is discontinuous at $x=0$ .