Chapter 2.4, Problem 68AYU

To determine

To find:

1. Check whether the equation $y$ is a function.

2. Domain and range of the function $y$ .

3. $x$ -intercepts and $y$ -intercepts if any.

4. Check whether the function is even, odd or neither.

5. Graph of $f$ .

Answer to Problem 68AYU

Solution:

1. Y is a function.

2. Domain is the set of all real numbers. Range $y=\{0,1\}$

3. The y-intercept is (0,1) and x-intercept is (All irrational numbers, 0)

4. The given function is an even.

5.  

  $f(x)=\{\begin{array}{ll}1,\hfill & \text{ if }x\text{ is rational }\hfill \\ 0,\hfill & \text{ if }x\text{ is irrational }\hfill \end{array}$

Explanation of Solution

Given:

It is given that the equation $y=\{\begin{array}{cc}1& \text{ if }x\text{ is rational }\\ 0& \text{ if }x\text{ is irrational }\end{array}$ and asked to find its domain, range, y-intercept, $x$ -intercelpt if any, nature of function and its graph.

since it takes all the real values as input and 0,1 as ouput given equation is function.

As $x$ takes all rational and irrational numbers, $x$ takes all the real numbers.

Domain is the set of all real numbers. Range $y=\{0,1\}$ . When $x$ takes $0, y$ must be 1 .

As 0 is a rational number, the corresponding $y$ must be 1. The $y$ -intercept is (0,1) .

As $x$ is irrational $y$ becomes 0 .

x-intercept is (All irrational numbers, 0)

As the output values lies in 0 and 1, the function must be symmetric about y-axis. Therefore

given function is an even.

To Sketch:

  

  $f(x)=\{\begin{array}{ll}1,\hfill & \text{ if }x\text{ is rational }\hfill \\ 0,\hfill & \text{ if }x\text{ is irrational }\hfill \end{array}$