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11.3: Exercises

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    Exercise \(\PageIndex{1}\)

    Find the domain, the vertical asymptotes and removable discontinuities of the functions.

    1. \(f(x)=\dfrac{2}{x-2}\)
    2. \(f(x)=\dfrac{x^2+2}{x^2-6x+8}\)
    3. \(f(x)=\dfrac{3x+6}{x^3-4x}\)
    4. \(f(x)=\dfrac{(x-2)(x+3)(x+4)}{(x-2)^2(x+3)(x-5)}\)
    5. \(f(x)=\dfrac{x-1}{x^3-1}\)
    6. \(f(x)=\dfrac{2}{x^3-2x^2-x+2}\)
    Answer
    1. domain \(D = \mathbb{R} − \{2\}\), vertical asymptote at \(x = 2\), no removable discontinuities
    2. \(D = \mathbb{R}−\{2, 4\}\), vertical asympt. at \(x = 2\) and \(x = 4\), no removable discont.
    3. \(D = \mathbb{R}− \{−2, 0, 2\}\), vertical asympt. at \(x = 0\) and \(x = 2\), removable discont. at \(x = −2\)
    4. \(D = \mathbb{R} − \{−3, 2, 5\}\), vertical asympt. at \(x = 2\) and \(x = 5\), removable discont. at \(x = −3\)
    5. \(D = \mathbb{R} − \{1\}\), no vertical asympt., removable discont. at \(x = 1\)
    6. \(D = \mathbb{R} − \{−1, 1, 2\}\), vertical asympt. at \(x = −1\) and \(x = 1\) and \(x = 2\), no removable discont.

    Exercise \(\PageIndex{2}\)

    Find the horizontal asymptotes of the functions.

    1. \(f(x)=\dfrac{8x^2+2x+1}{2x^2+3x-2}\)
    2. \(f(x)=\dfrac{1}{(x-3)^2}\)
    3. \(f(x)=\dfrac{x^2+3x+2}{x-1}\)
    4. \(f(x)=\dfrac{12x^3-4x+2}{-3x^3+2x^2+1}\)
    Answer
    1. \(y = 4\)
    2. \(y = 0\)
    3. no horizontal asymptote (asymptotic behavior \(y = x + 4\))
    4. \(y = −4\)

    Exercise \(\PageIndex{3}\)

    Find the \(x\)- and \(y\)-intercepts of the functions.

    1. \(f(x)=\dfrac{x-3}{x-1}\)
    2. \(f(x)=\dfrac{x^3-4x}{x^2-8x+15}\)
    3. \(f(x)=\dfrac{(x-3)(x-1)(x+4)}{(x-2)(x-5)}\)
    4. \(f(x)=\dfrac{x^2+5x+6}{x^2+2x}\)
    Answer
    1. \(x\)-intercept at \(x = 3\), \(y\)-intercept at \(y = 3\)
    2. \(x\)-intercepts at \(x = 0\) and \(x = −2\) and \(x = 2\), \(y\)-intercept at \(y = 0\)
    3. \(x\)-intercepts at \(x = −4\) and \(x = 1\) and \(x = 3\), \(y\)-intercept at \(y = \dfrac{6}{5}\)
    4. \(x\)-intercept at \(x = −3\) (but not at \(x = −2\) since \(f(−2)\) is undefined), no \(y\)-intercept since \(f(0)\) is undefined

    Exercise \(\PageIndex{4}\)

    Sketch the graph of the function \(f\) by using the domain of \(f\), the horizontal and vertical asymptotes, the removable singularities, the \(x\)- and \(y\)-intercepts of the function, together with a sketch of the graph obtained from the calculator.

    1. \(f(x)=\dfrac{6x-2}{2x+4}\)
    2. \(f(x)=\dfrac{x-3}{x^3-3x^2-6x+8}\)
    3. \(f(x)=\dfrac{x^4-10x^2+9}{x^2-3x+2}\)
    4. \(f(x)=\dfrac{x^3-3x^2-x+3}{x^3-2x^2}\)
    Answer
    1. \(D = \mathbb{R} − \{2\}\), horizontal asympt. \(y = 3\), vertical asympt. \(x = −2\), no removable discont., \(x\)-intercept at \(x = \dfrac 1 3\), \(y\)-intercept at \(y = \dfrac{-1}{2}\), graph:

    clipboard_ecbbba7ef054ee5a490309ff6ad0fbf4b.png

    1. \(f(x)=\dfrac{x-3}{(x-4)(x-1)(x+2)}\) has domain \(D = \mathbb{R} − \{−2, 1, 4\}\), horizontal asympt. \(y = 0\), vertical asympt. \(x = −2\) and \(x = 1\) and \(x = 4\), no removable discont., \(x\)-intercept at \(x = 3\), \(y\)-intercept at \(y = \dfrac{-3}{8} = −0.375\), graph:

    clipboard_e0d6db6e49d48f2bad98ab2b86fb86e1b.png

    1. \(f(x)=\dfrac{(x-3)(x+3)(x-1)(x+1)}{(x-2)(x-1)}\) has domain \(D = \mathbb{R} − \{1, 2\}\), no horizontal asympt., vertical asympt. \(x = 2\), removable discont. at \(x = 1\), \(x\)-intercept at \(x = −3\) and \(x = −1\) and \(x = 3\), \(y\)-intercept at \(y = \dfrac 9 2 = 4.5\), graph:

    clipboard_e3ce9e8190cd29dd861506874a33ab2d1.png

    1. \(f(x)=\dfrac{(x-3)(x-1)(x+1)}{x^{2}(x-2)}\) has domain \(D = \mathbb{R} − \{0, 2\}\), horizontal asympt. \(y = 1\), vertical asympt. \(x = 0\) and \(x = 2\), no removable discont., \(x\)-intercepts at \(x = −1\) and \(x = 1\) and \(x = 3\), no \(y\)-intercept since \(f(0)\) is undefined, graph:

    clipboard_e46938e7afbc22f2db2e88901bdcc36c6.png

    Note that the graph intersects the horizontal asymptote \(y = 1\) at approximately \(x \approx-2.3\) and approaches the asymptote from above

    Exercise \(\PageIndex{5}\)

    Find a rational function \(f\) that satisfies all the given properties.

    1. vertical asymptote at \(x=4\) and horizontal asymptote \(y=0\)
    2. vertical asymptotes at \(x=2\) and \(x=3\) and horizontal asymptote \(y=5\)
    3. removable singularity at \(x=1\) and no horizontal asymptote
    Answer
    1. for example \(f(x)=\dfrac{1}{x-4}\)
    2. for example \(f(x)=\dfrac{5 x^{2}}{x^{2}-5 x+6}\)
    3. for example \(f(x)=\dfrac{x^{2}-x}{x-1}\)

    This page titled 11.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.