Exercise 1.1.53 (53-58)

Question

Sketch the graph of the function.
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53. $f(x)=x+|x|$
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54. $f(x)=|x+2|$
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55. $g(t)=|1-3 t|$
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56. $f(x)=\frac{|x|}{x}$
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57. $f(x)= \begin{cases}|x| & 	ext { if }|x| \leq 1 \ 1 & 	ext { if }|x|>1\end{cases}$
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58. $g(x)=|| x|-1|$

Khagan Gulusoy · Accepted Answer

\begin{enumerate} \item [53.] The function $f$ can be equivalently rewritten as: $$f(x) = x+|x|=\left\{ \begin{array}{rcl} 0, & \text{ if } x\leq 0\ 2x, & \text { if } x>0. \end{array}\right.$$ Now we represent this function numerically (i.e. using the table of values): \begin{table}[H] \centering \begin{tabular}{|l|l|c|l|l|l|} \hline $x$ & -2 & -1 & 0 & 1 & 3 \ \hline $y$ & 0 & 0 & 0 & 2 & 6 \ \hline \end{tabular} \end{table} To get the graph, we only have to connect the points from the table: \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[axis lines=middle,xmin=-2.2,xmax=3.2,ymin=-0.2,ymax=6.2, xlabel={$x$}, ylabel={$f(x)$}, unit vector ratio=1 1 1,x label style = {at={(ticklabel cs:0.9)},anchor=north, above=6.7mm}, y label style={anchor=north, above=3mm}] \addplot[domain=-2.2:0, samples=2, mark=none, color=violet, ultra thick] {0}; \addplot[domain=0:3.2, samples=2, mark=none, color=violet, ultra thick] {2*x}; \addplot[mark=*, only marks] coordinates {(1,2) (3,6)}; \addplot[mark=*, only marks] coordinates {(-2,0) (-1, 0)}; \end{axis} \end{tikzpicture} \end{figure} \item[54.] We can rewrite $f$ as follows: $$f(x)=|x+2|=\left\{ \begin{array}{rcl} -(x+2), & \text{if }x+2<0 \ x+2, & \text{if } x+2\geq 0 \end{array}\right. = \left\{ \begin{array}{rcl} -x-2, & \text{if } x<-2\ x+2, & \text{if } x\geq -2 \end{array}\right.$$ Now we should represent $f$ numerically: \begin{table}[H] \centering \begin{tabular}{|*{5}{c|}} \hline $x$ & -4 & -2 & 0 & 1 \ \hline $f(x)$ & 2 & 0 & 2 & 3 \ \hline \end{tabular} \end{table} Now we are able to sketch the graph of the function $f$: \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[axis lines=middle,xmin=-4.2,xmax=3.2,ymin=-0.2,ymax=5.2, xlabel={$x$}, ylabel={$f(x)$}, unit vector ratio=1 1 1,x label style = {at={(ticklabel cs:0.9)},anchor=north, above=6.7mm}, y label style={anchor=north, above=3mm}] \addplot[domain=-4.2:-2, samples=2, mark=none, color=black!50!green, ultra thick] {-x-2}; \addplot[domain=-2:3.2, samples=2, mark=none, color=black!50!green, ultra thick] {x+2}; \addplot[mark=*, only marks] coordinates {(1,3) (0,2)}; \addplot[mark=*, only marks] coordinates {(-2,0) (-4, 2)}; \end{axis} \end{tikzpicture} \end{figure} \item [55.] We can rewrite $g$ as a piecewise defined function: $$g(t)=|1-3t|=\left\{ \begin{array}{rcl} 1-3t, & \text{ if } t\leq \frac{1}{3}\ 3t-1, & \text{ if } t>\frac{1}{3} \end{array}\right.$$ Now we should represent $g$ numerically: \begin{table}[H] \centering \begin{tabular}{|*{6}{c|}} \hline $x$ & -2 & -1 & 0 & 1 & 3 \ \hline $y$ & 7 & 4 & 1 & 2 & 8 \ \hline \end{tabular} \end{table} We can sketch the graph of $g$ by connecting points in the table: \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[axis lines=middle,xmin=-2.2,xmax=3.2,ymin=-0.2,ymax=8.2, xtick={-2,-1,1/3,1,2,3} ,unit vector ratio=1 1 1,xticklabels={-2,-1,$\frac{1}{3}$, 1,2,3},xlabel={$t$}, ylabel={$g(t)$}, x label style = {at={(ticklabel cs:0.9)},anchor=north, above=6.7mm}, y label style={anchor=north, above=3mm}] \addplot[domain=-2.2:1/3, samples=2, mark=none, color=violet, ultra thick] {1-3*x}; \addplot[domain=1/3:3.2, samples=2, mark=none, color=violet, ultra thick] {3*x-1}; \addplot[mark=*, only marks] coordinates {(1,2) (3,8)}; \addplot[mark=*, only marks] coordinates {(-2,7) (-1, 4) (0,1)}; \end{axis} \end{tikzpicture} \end{figure} \item[56.] We can rewrite the function $f$ as follows: $$f(x)=\frac{|x|}{x}=\left\{ \begin{array}{rcl} 1, & \text{ if }x>0\ -1, & \text{ if } x<0 \end{array}\right.$$ Note that $x\neq 0$ (since the denominator $x$ cannot be equal to zero). Thus, the function $f$ is defined on the interval $(-\infty, 0)\cup (0, +\infty)$. We sketch the graph of the function $f$ as follows: \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[axis lines=middle,xmin=-2.2,xmax=2.2,unit vector ratio=1 1 1,ymin=-2,ymax=2,xlabel={$x$}, 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=-2.2:0, samples=2, mark=none, color=blue, ultra thick] {-1}; \addplot[domain=0:2.2, samples=2, mark=none, color=blue, ultra thick] {1}; \addplot[mark=*, only marks, fill=white] coordinates {(0,1) (0,-1)}; \end{axis} \end{tikzpicture} \end{figure} \item[57.] Let's reformulate the definition of the function $f$ and then represent it numerically: $$f(x)= \left\{ \begin{array}{rcl} |x|, &\text{ if }|x|\leq 1\ 1, &\text{ if }|x|>1 \end{array}\right. = \left\{ \begin{array}{rcl} 1, &\text{ if }x< -1\ |x|,&\text{ if }-1\leq x\leq 1\ 1, &\text{ if }1< x \end{array}\right. = \left\{ \begin{array}{rcl} 1, &\text{ if }x< -1\ -x, &\text{ if }-1\leq x\leq 0\ x, &\text{ if }0< x\leq 1\ 1, &\text{ if }1< x \end{array}\right.$$ \begin{table}[H]\centering \begin{tabular}{|l|l|c|l|l|l|l|l|} \hline $x$ & -3 & -2 & -1 & 0 & 0.5 & 1 & 2 \ \hline $f(x)$ & 1 & 1 & 1 & 0 & 0.5 & 1 & 1 \ \hline \end{tabular} \end{table} After we connect the points, we get: \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[axis lines=middle, unit vector ratio=1 1 1, xmin=-2.2, xmax=2.2, ymin=-0.2, ymax=1.2, xlabel={$x$}, 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:-1, samples=2, mark=none, color=black!30!green, ultra thick] {1}; \addplot[domain=1:2.2, samples=2, mark=none, color=black!30!green, ultra thick] {1}; \addplot[domain=-1:0, samples=50, mark=none, color=black!30!green, ultra thick] {-x}; \addplot[domain=0:1, samples=50, mark=none, color=black!30!green, ultra thick] {x}; \end{axis} \end{tikzpicture} \end{figure} \item[58.] \begin{align*} g(x)=||x|-1| &= \left\{ \begin{array}{ll} |x|-1, &\text{if }\, |x|-1\geq 0\ 1-|x|, &\text{if }\, |x|-1<0 \end{array}\right. = \left\{ \begin{array}{ll} |x|-1, &\text{ if } |x|\geq 1\ 1-|x|, &\text{ if } |x|<1 \end{array}\right.\[10pt] &= \left\{ \begin{array}{ll} |x|-1, &\text{ if } x\leq -1\ 1-|x|, &\text{ if } -1< x\leq 1\ |x|-1, &\text{ if } 1< x \end{array}\right. =\left\{ \begin{array}{ll} -x-1, &\text{ if } x\leq -1\ 1+x, &\text{ if } -1< x\leq 0\ 1-x, &\text{ if } 0< x< 1\ x-1, &\text{ if } x\geq 1 \end{array}\right. \end{align*} Now we represent this function numerically: \begin{table}[H] \centering \begin{tabular}{|*{6}{c|}} \hline $x$ & -3 & -2 & 0 & 1 & 2 \ \hline $f(x)$ & 2 & 1 & 1 & 0 & 1 \ \hline \end{tabular} \end{table} Now we can sketch the graph of $g:$ \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[axis lines=middle, unit vector ratio=1 1 1, xmin=-2.2, xmax=2.2, ymin=-0.2, ymax=1.2, xlabel={$x$}, 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:-1, samples=50, mark=none, color=black!30!green, ultra thick] {abs(x)-1}; \addplot[domain=-1:1, samples=50, mark=none, color=black!30!green, ultra thick] {1-abs(x)}; \addplot[domain=1:2.2, samples=50, mark=none, color=black!30!green, ultra thick] {abs(x)-1}; \end{axis} \end{tikzpicture} \end{figure} \end{enumerate}

Exercise 1.1.53 (53-58)

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$x$	-4	-2	0	1

$f (x)$	2	0	2	3


$x$	-3	-2	0	1	2

$f (x)$	2	1	1	0	1

Exercise 1.1.53 (53-58)

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