Q. 40 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 29–40:  

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

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

Step 1. Given Information

We are given a function,

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

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

The domain of function 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 x- intercept is $(x,0)$ for $0\le x<\frac{1}{2}$ and there is no y- intercept.

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 set of all integers. 

Part(e) Step 1. Checking continuity

As seen from the graph, the graph is discontinuous at every multiple of $\frac{1}{2}$ , so the function f  is not continuous at  $\left\{x\right|x \text{is an integral multiple} \text{of} \frac{1}{2}\}$