Exercise 1.1.48

Question

Find the domain and sketch the graph of the function:
    $$f(x)=\frac{x^2-4}{x-2}$$

Khagan Gulusoy · Accepted Answer

\begin{itemize}
    \item 	extit{(domain)} Since the denominator $x-2$ cannot be equal to zero, we have:
    \begin{align*}
        \forall x \in \mathbb{R}:\quad 
        &x-2
eq 0\
        &\iff x
eq 2.
    \end{align*}
    In other words, the domain of $f$ is $\{x \in \mathbb{R} \mid x 
eq 2\} = (-\infty, 2)\cup (2, +\infty)$. 
    \item 	extit{(graph)} To sketch the graph of the function $f$, we first need to simplify it:
    $$\forall x \in (-\infty,2)\cup(2,+\infty): \quad f(x)=\frac{x^2-4}{x-2}=\frac{(x-2)(x+2)}{x-2}=x+2.$$
    Since the function $f$ seems to be a linear function, we have to find only two points that belong to the graph:
    \begin{table}[H]\centering
\begin{tabular}{|l|l|l|}
\hline
$x$ & $-1$ & $1$ \ \hline
$y$ & $1$  & $3$ \ \hline
\end{tabular}
\end{table}
    We can now easily sketch the graph of the function $f$:
    \begin{figure}[H] \centering
\begin{tikzpicture}
    \begin{axis}[axis lines=middle,width=6cm, height=6cm,xmin=-2.2,xmax=2.2,ymin=-1,ymax=5,xlabel={$x$}, grid=both,
    grid style={line width=.3pt, draw=black!10}, ylabel={$f(x)$}, x label style = {at={(ticklabel cs:0.9)},anchor=north, above=6.7mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-2.2:2.2, samples=2, mark=none, color=violet, ultra thick] {x+2}; 
         \addplot[mark=*, only marks] coordinates {(-1, 1) (1,3)}; 
         \addplot[mark=*, fill=white, only marks] coordinates {(2, 4)}; 
         
    \end{axis}
\end{tikzpicture}
\end{figure}
\end{itemize}

Exercise 1.1.48

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Exercise 1.1.48

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