Question

Consider the equation

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

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?

Short answer

The equation $y=\left\{\begin{array}{l}1if\mathit{}x\mathit{}is\mathit{}rational\\ 0ifxisirrational\end{array}\right.$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-by-step solution

Step 1. Given information

For the equation $y=\left\{\begin{array}{l}1if\mathit{}x\mathit{}is\mathit{}rational\\ 0ifxisirrational\end{array}\right.$, 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.