Chapter 1.2, Problem 71E

Solution

To find: the given statement is true or false.

True.

Given information: 

The graph of function $f$ is defined as the set of all points $(x,f(x))$ , where $x$ is in the

domain of $f$ .

Calculation:

The simplest explanation here is that the statement is true because that's the very definition of a graph is any connection (not just functions), a graph is the set of points $\left(a,b\right)$ of each $a$ in the domain that is mapped to a $b$ in the range.

In a function $f$ , each $x$ in the domain is mapped to a $f\left(x\right)$ in the range. So, any point in the graph of $f$ will necessarily be $(x,f(x))$ for any $x$ in the domain.

For example, if $f$ is defined as

  $f\left(x\right)=\left\{\begin{array}{l}1,x=0\hfill \\ 3,x=2\hfill \\ 10,x=1\hfill \end{array}\right.$

Then the graph of $f$ is

  $\{(0,1),(2,3),(1,10)\}$

Note that a graph is not necessarily a drawing on a Cartesian plane. One approach to depict a graph is on a coordinate plane. Another is to write the graph as a set.

It becomes incredibly difficult to list the points one by one because the majority of functions deal with are defined for considerably larger sets, all real numbers, which have arbitrarily huge domains.

However, one may still show the graph of, say, $f\left(x\right)=x^2$ .

  $\left(x,f(x)\right)$ such that $f\left(x\right)=x^2$ for all real $x$ .

Despite the fact that for the majority of applications, a graphical representation on the Cartesian plane is considerably more useful.