 # Find the domain and range of these functions. • the function that assigns to each pair of positive integers the first integer of the pair.
• the function that assigns to each positive integer its largest decimal digit.
• the function that assigns to a bit string the number of ones minus the number of zeros in the string.
• the function that assigns to each positive integer the largest integer not exceeding the square root of the integer.
• the function that assigns to a bit string the longest string of ones in the string.

The objective of this question is to find the domain and range of the given functions.

A relation’s domain and range are characterized as a set containing all the $x$ and $y$ coordinates of the ordered pairs, respectively. Similarly, the range and domain of a function are the elements of a function. A set containing all the potential values that a function can possess is said to be the domain of a function. In other words, we can say that a domain is a set incorporating all the possible values that an independent variable can possess. It is easy to find the domain of a rational function if the denominator does not equate to zero as well as if the function has all the real values.

To determine the domain, we must examine the values of the independent variables that are permitted to be used. That is, the denominator of a fraction should not be $0$ and a positive number in the square root. The set containing all the potential outputs of a function is said to be its range, in other words, the values possesses by a dependent variable when the values of independent variables are plugged into the function. Let $y=f(x)$ be a function, its range will be the expansion of $y$ values from lowest to highest.

Let $D$ be the domain and $R$ be the range of the given functions.

For the first set, the domain incorporates a set of all the positive integer pairs excluding a non-positive natural number zero. Mathematically, it refers to $N-\{0\}$, hence:

$D=\{(x,y)| x= 1,2,3,\cdots\text{ and } y= 1,2,3,\cdots\}$

$=\{(x,y)| x\in N-\{0\}\wedge y\in N-\{0\}\}$

Or $D=(N-\{0\})\times (N-\{0\})$

And $R=\{1,2,3,\cdots\}=N-\{0\}$

For the second function:

$D=\{1,2,3,\cdots\}=N-\{0\}$

And its range will be values from $1$ to $9$:

$R=\{1,2,3,4,5,6,7,8,9\}$

For the third function, the domain will be a set of all the bit strings as:

$D=\{\lambda,0,1,00,01,11,10,010,011,\cdots\}$

The range may possess negative, positive, or zero values, being a set containing the difference of several $1$’s and $0$’s in a string.  So the range will be:

$R=\{\cdots,-2,-1,0,1,2,3,\cdots\}$

For the fourth function:

$D=\{1,2,3,\cdots\}=N-\{0\}$

We can see that the set contains all positive integers, so:

$R=\{1,2,3,\cdots\}=N-\{0\}$

For the last function, the domain will be a set of all the bit strings as:

$D=\{\lambda,0,1,00,01,11,10,010,\cdots\}$

And the range will be a set containing the string having only the digit $1$ as:

$R=\{\lambda,1,11,111,1111,11111,\cdots\}$

## Example

Find both the domain and range of the function $f(x)=x^2-2$.

### Solution

Since the function will be defined for all values of $x$, so we have the domain:

$D=(-\infty,\infty)$

To find the range:

Let  $y=x^2-2$

or  $x=\sqrt{y+2}$

$\sqrt{y+2}$ will be defined for $y+2\geq 0$  or  $y\geq -2$

Thus, the range will be:

$[-2,\infty)$