Q. 29 - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=\left\{\begin{array}{l}2x if x\ne 0\\ 1 if x=0\end{array}\right.$

(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 given function is the set of all real numbers.

(b)  $(0,1)$ is a point on the graph.

(c)   Graph of the function

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

(e) The function f is discontinuous on its domain.

Step 1.Given information

The given function $f\left(x\right)=\left\{\begin{array}{l}2x if x\ne 0\\ 1 if x=0\end{array}\right.$

Step 2.Find the domain of each function

The object is to find the domain of the function

The domain of the function $f\left(x\right)$,is the set of all the possible values of x

The value of the function $f\left(x\right)$ at x=0 is 1.So,$f\left(0\right)=1$ . The value of function for any value of x,other than 0 is given by .

In the expression $2x$ ,The value of the variable x is multiplied by .These operations can be performed on any real number.

So,the domain of x is the set of all real numbers.

Therefore,the domain of x the given function is the set of all real numbers.

Step 3.Locate any intercepts

The intercepts is to locate any intercepts of function .

The $y$ - intercepts are those points  for which the   x- coordinate is zero.

The value of the function $f\left(x\right)$ or the y-coordinate  is 1.When x=0.

So,the y– intercept is 1.Therefore,$(0,1)$ is a point on the graph.

Step 4.Graph each function

The objective is to graph the function.

Graph each piece to graph the function.

For platting the graph of the line $y=2x$, substitute various values for x in the equation to obtain some points on the graph.

$x$	$y=2x$	$(x,y)$
1	2	$(1,2)$
-1	-2	$(-1,-2)$

The point $(0,1)$ is also on the graph because x=0,$f\left(x\right)=1.$

Plot the points and draw the line to get the graph of the function.

Step 5.Based on the graph,Find the range

The object is to find the range of f,based on its graph.

From,the graph notice that the points on the graph of f have the y-coordinates between 3 and -3,excluding 0.For each number y,there is atleast one number x in the domain.

So,the range of f is $\left\{y|y\ne 0\right\}$ or the interval $(-\infty ,0)\cup (0,\infty ).$

Step 6.Checking f continuous on its domain?

The objective is to determine wheather the function f is continuous or not.

From,the graph,it can be observed that there is discontinuity at x=0.So,the function f is discontinuous on its domain.