3.7e: Exercises for the reciprocal function
- Page ID
- 44351
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A: Graph translations of \( \dfrac{1}{x} \)
Exercise \(\PageIndex{A}\)
\( \bigstar \) Graph the given function. Identify the translations on \(y = \dfrac{1}{x}\) used to sketch the graph. Then state the domain and range.
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- Answers to odd exercises:
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1. Shift right \(2\) units;
domain: \((−∞, 2) ∪ (2, ∞)\);
range: \((−∞, 0) ∪ (0, ∞)\)
3. Shift up \(5\) units;
domain: \((−∞, 0) ∪ (0, ∞)\);
range: \((−∞, 1) ∪ (1, ∞)\)
5. Shift left \(1\) unit and down \(2\) units;
domain: \((−∞, −1) ∪ (−1, ∞)\);
range: \((−∞, −2) ∪ (−2, ∞)\)
\( \bigstar \) Use the transformations to graph the following functions.
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- Answers to odd exercises:
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7. Reflect over \(x\)-axis;
domain: \((−∞, 0) ∪ (0, ∞)\);
range: \((−∞, 0) ∪ (0, ∞)\)
9. Shift left\(2\) units,
Reflect over \(x\)-axis;
domain: \((−∞, -2) ∪ (-2, ∞)\);
range: \((−∞, 0) ∪ (0, ∞)\)
11. Shift left\(2\) units,
Reflect over \(y\)-axis;
domain: \((−∞, -2) ∪ (-2, ∞)\);
range: \((−∞, 0) ∪ (0, ∞)\)
\( \bigstar \) Graph using translations of \( \dfrac{1}{x} \) by first using division to rewrite the function.
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- Answers to odd exercises:
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13. \(f(x) = 4+\frac{1}{x-2}\)
Right 2, up 4
15. \(f(x) = -2+\frac{1}{x+5}\)
Left 5, down 2
17. \(f(x) = -4 - \frac{1}{x-1}\)
Right 1, reflect over x-axis, down 4
\( \bigstar \) Graph using translations of \( \dfrac{1}{x} \) by first using division to rewrite the function.
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- Answers to odd exercises:
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19. \(f(x) = 1-5\frac{1}{x+2}\)
Left 2, Reflect over x-axis, y -> 5y, up 1
21. \(f(x) = -1+6\frac{1}{x+1}\)
Left 1, y -> 6y, down 1
23. \(f(x) = 2 - 7 \frac{1}{x+1}\)
Left 1, reflect over x-axis, y -> 7y, up 2
B: Construct a graph from a verbal description
Exercise \(\PageIndex{B}\)
\( \bigstar \) Use the given transformation to graph the function. Note the vertical and horizontal asymptotes.
- The reciprocal function shifted up two units.
- The reciprocal function shifted down one unit and left three units.
- The reciprocal squared function shifted to the right \(2\) units.
- The reciprocal squared function shifted down \(2\) units and right \(1\) unit.
- Answers to odd exercises.
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31. V.A. \(x=0\), H.A. \(y=2\)
33. V.A. \(x=2\), H.A. \(y=0\)
\( \bigstar \)