Q. 39 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 29–40:  

$f\left(x\right)=2int\left(x\right)$

1. Find the domain of each function.
2. Locate any intercepts.
3. Graph each function.
4. Based on the graph, Find the range.
5. Is f continuous on its domain?

Step-by-Step Solution

Answer

1. The domain of the function is all real numbers. .
2. The x- intercepts is $(x,0)$ for $0\le x<1$.
3. The graph of the function is as follows,
4. The range of the function is the set of even integers.
5. The function f  is not continuous 

Step 1. Given Information

We are given a function,

$f\left(x\right)=2int\left(x\right)$

Part(a) Step 1. Finding the domain of the function

The domain of function $f\left(x\right)=2int\left(x\right)$ is all real numbers.

Part(b) Step 1. Locating intercepts

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

The value of function is zero when $0\le x<1$

So. the x- intercept is $(x,0)$ for $0\le x<1$

Part(c) Step 1. Graphing the function

The table of values for graphing the function is as follows, 

The graph is as follows,

Part(d) Step 1. Finding the range

As seen from the graph, the range of the function is the set of even integers.

Part(e) Step 1. Checking continuity

The function f  is not continuous because it is discontinuous at $\left\{x\right|x is an integer\}$