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Section 7.2 Graphs of Functions
In
Definition 7.2 we had introduced the image of an element of the domain under a function. The set of all these images is called the image of the function.
Definition 7.13 . Image of a function.
\(f:A\to B\)
\begin{equation*}
f(A) = \left\{ f(x) \mid x \in A \right\}.
\end{equation*}
For all
\(x\) in the domain of
\(f:A\to B\) we have
\(f(x)\in B\text{,}\) thus the image
\(f(A)\) of
\(f\) is a subset of codomain
\(B\) of
\(f\text{.}\)
Example 7.14 .
Let \(f:\mathbb{Z}_5\to \mathbb{Z}_5\) be given by \(f(x)=x^2\bmod 5\text{.}\) Then
\begin{gather*}
f(0)=0^2\bmod 5=0\bmod 5 = 0\\
f(1)=1^2\bmod 5=1\bmod 5 = 1\\
f(2)=2^2\bmod 5=4\bmod 5 = 4\\
f(3)=3^2\bmod 5=9\bmod 5 = 4\\
f(4)=4^2\bmod 5=16\bmod 5 = 1
\end{gather*}
Thus the image \(f(\mathbb{Z}_5)\) of \(f\) is
\begin{equation*}
f(\mathbb{Z}_5)=\left\{ f(x) \mid x \in \mathbb{Z}_5 \right\}=\{0,1,4\}.
\end{equation*}
Note that this differs from the codomain of \(f\) which is \(\mathbb{Z}_5=\{0,1,2,3,4\}\text{.}\)
In
Section 7.1 we defined functions by algebraic expressions or tables of charts. Another way of representing a function is its graph. Instead of organizing the values in a table we consider them as elements of a Cartesian product. We give an introduction to graphs of functions in the video in
Figure 7.15 . It is followed by a more detailed discussion.
Figure 7.15. Graphs of Functions by Matt Farmer
Definition 7.16 . Graph of a function.
\(f:A\to B\)
\begin{equation*}
\left\{ (x,f(x)) \mid x \in B \right\} \subseteq A\times B.
\end{equation*}
In
Example 7.17 we obtain a graphical representation a function by applying the graphical representation of Cartesian products from
Section 6.3 to the graph of the function.
Example 7.17 .
Let
\(f:\mathbb{Z}_{6}\to\mathbb{Z}_{5}\) be given by
\(f(x)= 2^x \bmod 5\text{.}\)
We have
\begin{align*}
f(0) \amp= 2^0 \bmod 5 = 1 \bmod 5 = 1\\
f(1) \amp= 2^1 \bmod 5 = 2 \bmod 5 = 2\\
f(2) \amp= 2^2 \bmod 5 = 4 \bmod 5 = 4\\
f(3) \amp= 2^3 \bmod 5 = 8 \bmod 5 = 3\\
f(4) \amp= 2^4 \bmod 5 = 16 \bmod 5 = 1\\
f(5) \amp= 2^5 \bmod 5 = 32 \bmod 5 = 2
\end{align*}
Thus the graph of
\(f\) is:
\begin{align*}
\amp\{(x,f(x))\mid x\in \Z_6\}\\
\amp\;=\{(0,f(0)), (1,f(1)), (2,f(2)), (3,f(3)), (4,f(4)), (5,f(5))\}\\
\amp\;=\{(0,1), (1,2), (2,4), (3,3), (4,1), (5,2)\} \subseteq \mathbb{Z}_{6}\times\mathbb{Z}_{5}
\end{align*}
The graph of
\(f\) and
Section 6.3 yield a graphical representation of the function
\(f\text{.}\)
The graphical representation of a function yields not only yields its graph but also its domain and codomain, as illustrate in
Example 7.18 .
Example 7.18 .
Suppose that the graph of the function
\(h\) is given by
The values on the horizontal axis of the plot are the elements of domain of \(h\text{.}\) So the domain of is
\begin{equation*}
\mathbb{Z}_{10}=\{0,1,2,3,4,5,6,7,8,9\}\text{.}
\end{equation*}
The codomain are the values on the vertical axis of the plot. Thus the codomain of \(h\) is
\begin{equation*}
\mathbb{Z}_{5}=\{0,1,2,3,4\}\text{.}
\end{equation*}
The graph of \(h\) are the elements of the Cartesian product \(\mathbb{Z}_{10}\times\mathbb{Z}_5\) which are represented by the black pixels in the plot. We find that the graph of \(h\) is
\begin{equation*}
\{ (x,h(x)) \mid x \in A \} = \{
(0,2),
(1,0),(2,0),
(3,1),
(4,0),
(5,1),
(6,0),
(7,4),(8,4),(9,3)
\}.
\end{equation*}
Because the graph of \(h\) consists of the pairs \((x,h(x))\) where \(x\) is an element of the domain of \(h\text{,}\) we can read off the values \(h(x)\) easily.
In
Example 7.19 observe how changing the plot of the graph of a function changes its graph.
Example 7.19 .