Q. 68 - Precalculus Enhanced with Graphing Utilities

Question

Consider the equation 

$y=\left\{\begin{array}{l}1 if\mathit{ }x\mathit{ }is\mathit{ }rational\\ 0 if x is irrational\end{array}\right.$

Is this a function? What is its domain? What is its range? What is its y-intercept, if any? What are its x-intercepts, if any? Is it even, odd, or neither? How would you describe its graph? 

Step-by-Step Solution

Answer

The equation $y=\left\{\begin{array}{l}1 if\mathit{ }x\mathit{ }is\mathit{ }rational\\ 0 if x is irrational\end{array}\right.$is a function.

The domain is the set of all real numbers and its range is given by the set $\left\{0,1\right\}$.

The y-intercept of the function is $(0,1)$ and the function has infinite x intercepts.

The function is even.

The graph of the function is discrete.

Step 1. Given information

For the equation $y=\left\{\begin{array}{l}1 if\mathit{ }x\mathit{ }is\mathit{ }rational\\ 0 if x is irrational\end{array}\right.$, we can see that any value of $x$ corresponds to only one value of $y$.

So the given equation is a function.

Step 2. Domain and Range

The function can take any $x$ rational or irrational.

So the domain of the function is the set of all real numbers.

The function has only two outputs $0$ and $1$. So its range is given by the set $\left\{0,1\right\}$

Step 3. Identify the intercepts

As $0$ is a rational number so when $0$ is the input of the function the output is $1$. So the point $(0,1)$ lies on the function. So the $y$ intercept of the function is $(0,1)$.

For every input which is irrational number the output that is the $y$ coordinate is $0$.

So the function has infinte $x$ intercepts.

Step 4. Describe the graph

Putting negative values of $x$ does not change the output of the function because if $x$ is rational then $-x$ is also rational and if $x$ is irrational then $-x$ is also irrational.

So $f\left(x\right)=f(-x)$ for all $x$ and hence the function is even.

The graph of the function is discrete giving the output $0$ when the input is irrational and giving the output $1$ when the input is rational. There will be only lots of points and the graph would be discontinuous in nature.