Skip to main content
Mathematics LibreTexts

3.7e: Exercises for the reciprocal function

  • Page ID
    44351
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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.

    1. \(f(x) = \dfrac{1}{x−2}\)
    2. \(f(x) = \dfrac{1}{x+3}\)
    1. \(f(x) = \dfrac{1}{x} + 5\)
    2. \(f(x) = \dfrac{1}{x} − 3\)
    1. \(f(x) = \dfrac{1}{x+1} − 2\)
    2. \(f(x) = \dfrac{1}{x−3} + 4\)
    Answers to odd exercises:

    1. Shift right \(2\) units;
    domain: \((−∞, 2) ∪ (2, ∞)\);
    range: \((−∞, 0) ∪ (0, ∞)\)

    75fa23d883d738eeb47a020057002b8f.png

    3. Shift up \(5\) units;
    domain: \((−∞, 0) ∪ (0, ∞)\);
    range: \((−∞, 1) ∪ (1, ∞)\)

    53d3a12d61be06d8913ae13668760ebb.png

    5. Shift left \(1\) unit and down \(2\) units;
    domain: \((−∞, −1) ∪ (−1, ∞)\);
    range: \((−∞, −2) ∪ (−2, ∞)\)

    0eac4ad67881e57bfa8e7dc46c933e8e.png

    \( \bigstar \) Use the transformations to graph the following functions.

    1. \(f ( x ) = - \dfrac { 1 } { x }\)
    2. \(f ( x ) =  \dfrac { -1 } { x + 1 } + 2\)
    1. \(f ( x ) =  \dfrac {- 1 } { x + 2 }\)
    2. \(f ( x ) =  \dfrac { 1 } {- x }\)
    1. \(f ( x ) =  \dfrac { 1 } {- x + 2 }\)
    2. \(f ( x ) =  \dfrac { 1 } { -x - 1 } + 2\)
    Answers to odd exercises:

    7. Reflect over \(x\)-axis;
    domain: \((−∞, 0) ∪ (0, ∞)\);
    range: \((−∞, 0) ∪ (0, ∞)\)

    3.7E #7.png

    9. Shift left\(2\) units,
    Reflect over \(x\)-axis;
    domain: \((−∞, -2) ∪ (-2, ∞)\);
    range: \((−∞, 0) ∪ (0, ∞)\)

    3.7E  2 2.png

    11. Shift left\(2\) units,
    Reflect over \(y\)-axis;
    domain: \((−∞, -2) ∪ (-2, ∞)\);
    range: \((−∞, 0) ∪ (0, ∞)\)

    3.7E #11.png

    \( \bigstar \) Graph using translations of \( \dfrac{1}{x} \) by first using division to rewrite the function.

    1. \(f(x) = \dfrac{4x-7}{x-2} \\[6pt]\)
    2. \(f(x) = \dfrac{4-x}{x−3} \\[6pt]\)
    1. \(f(x) = \dfrac{-2x-9}{x+5} \\[6pt]\)
    2. \(f(x) = \dfrac{2x+5}{x+3} \\[6pt]\)
    1. \(f(x) = \dfrac{3-4x}{x-1} \\[6pt]\)
    2. \(f(x) = \dfrac{x+1}{x+2} \\[6pt]\)
    Answers to odd exercises:

    13. \(f(x) = 4+\frac{1}{x-2}\)

    Right 2, up 4

    3.7E #13.png

    15. \(f(x) = -2+\frac{1}{x+5}\)

    Left 5,  down 2

    3.7E #15.png

    17. \(f(x) = -4 -  \frac{1}{x-1}\)

    Right 1, reflect over x-axis, down 4

    3.7E #17.png

    \( \bigstar \) Graph using translations of \( \dfrac{1}{x} \) by first using division to rewrite the function.

    1. \(f(x) = \dfrac{x−3}{x+2} \\[6pt]\)
    2. \(f(x) = \dfrac{x+2}{x−4} \\[6pt]\)
    1. \(f(x) = \dfrac{5−x}{x+1} \\[6pt]\)
    2. \(f(x) = \dfrac{x+2}{4−x} \\[6pt]\)
    1. \(f(x) = \dfrac{2x−5}{x+1} \\[6pt]\)
    2. \(f(x) = \dfrac{2x+5}{3−x} \\[6pt]\)
    Answers to odd exercises:

    19. \(f(x) = 1-5\frac{1}{x+2}\)

    Left 2, Reflect over x-axis, y -> 5y, up 1

    Screen Shot 2019-08-19 at 1.33.18 PM.png

    21. \(f(x) = -1+6\frac{1}{x+1}\)

    Left 1, y -> 6y, down 1

    Screen Shot 2019-08-19 at 1.33.56 PM.png

    23. \(f(x) = 2 - 7 \frac{1}{x+1}\)

    Left 1, reflect over x-axis, y -> 7y, up 2

    Screen Shot 2019-08-19 at 1.34.34 PM.png

    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.

    1. The reciprocal function shifted up two units.
    2. The reciprocal function shifted down one unit and left three units.
    3. The reciprocal squared function shifted to the right \(2\) units.
    4. The reciprocal squared function shifted down \(2\) units and right \(1\) unit.
    Answers to odd exercises.

    31. V.A. \(x=0\), H.A. \(y=2\)

    3.7e #31.png 

    33. V.A. \(x=2\), H.A. \(y=0\)

    CNX_Precalc_Figure_03_07_203.jpg 

     

    \( \bigstar \)


    3.7e: Exercises for the reciprocal function is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?