Chapter 2, Problem 1CRQ

To determine

To fill: If $f$ is a function from set $A$ to the set $B$, then $A$ is called the _____ of $f$, and the set of all values of $f\left(x\right)$ as $x$ takes on all possible values in $A$ is called the ______ of $f$. The range of $f$ is contained in the set ___.

Answer to Problem 1CRQ

If $f$ is a function from set $A$ to the set $B$, then $A$ is called the domain of $f$, and the set of all values of $f\left(x\right)$ as $x$ takes on all possible values in $A$ is called the range of $f$. The range of $f$ is contained in the set $B$.

Explanation of Solution

The notation $f:x\to y$ means that $f$ is a function from $x$ to $y$ where $x$ is called the domain of the function $f$ and $y$ is called the co-domain of the function $f$.

Let the unique element $y$ to which $f$ relates $x$ is denoted by $y=f\left(x\right)$. The set of all values of $f\left(x\right)$ taken together is called the range of $f$ and the range of $f$ in contained in $y$.

Example,

$R=\left\{\left(x,y\right):y=x+6\text{where}x,y\in N\text{and}x<3\right\}$

Substitute $1$ for $x$ in the above relation to find the value of $y$,

$\begin{array}{c}y=1+6\\ =7\in N\end{array}$

Substitute $1$ for $x$ in the above relation to find the value of $y$,

$\begin{array}{c}y=2+6\\ =8\in N\end{array}$

Hence, $R=\left\{\left(1,7\right)\left(2,8\right)\right\}$ where domain of $R$ is $\left\{1,2\right\}$ and range of $R$ is $\left\{7,8\right\}$.

Thus, if $f$ is a function from set $A$ to the set $B$, then $A$ is called the domain of $f$, and the set of all values of $f\left(x\right)$ as $x$ takes on all possible values in $A$ is called the range of $f$. The range of $f$ is contained in the set $B$.