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Mathematics LibreTexts

3.5: Rational Functions

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    1. Rational Functions (Definition)

    Definition: Rational Function

    A rational function is a quotient of polynomials \(\dfrac{P(x)}{Q(x)}\).

    Example 1

    \[\dfrac{(x^2 + x - 1)}{(3x^3+ 1)},\]

    \[\dfrac{(x - 1)}{(x^2 +1)}, \text{ and}\]

    \[\dfrac{x^2}{(x + 1)}\]

    are all Rational Functions

    Example 2

    Find the domain of

    \[\dfrac{(x^2 + 1)}{(x^2 -1)}.\]

    Solution

    The domain of this rational function is the set of all real numbers that do not make the denominator zero. We find

    \[x^2 -1 = 0\]

    solving

    \[x = 1, \;\;\; \text{or} \;\;\; x = -1.\]

    So that the domain is

    \[\{x | x \text{ is not }1 \text{ or } -1\}.\]

    2. Vertical Asymptotes

    Definition: Vertical Asymptote

    A Vertical Asymptote of a rational function occurs where the denominator is 0.

    Example 3

    Graph the vertical asymptotes of

    \[\dfrac{(x^2 + 1)}{(x^2 -1)}\]

    Solution

    From the last example, we see that there are vertical asymptotes at 1 and -1.

    Since \(f(x)\) is positive a little to the left of -1, we say that as

    \[x \rightarrow -1^{-} \text{ ("x goes to -1 from the left")},\]

    \[f(x) \rightarrow \infty.\]

    Similarly since \(f(x)\) is negative a little to the right of -1, we say that as

    \[x\rightarrow -1^{+} \text{( "x goes to -1 from the right")}, \]

    \[f(x) \rightarrow -\infty.\]

    Since \(f(x)\) is negative a little to the left of 1, as

    \[x \rightarrow1^{-},\]

    \[f(x) \rightarrow -\infty.\]

    Similarly since \(f(x)\) is positive a little to the right of 1, as

    \[x\rightarrow1^{+},\]

    \[f(x) \rightarrow\infty.\]

    Four Types of Vertical Asymptotes

    Below are the four types of vertical asymptotes:

    3. Horizontal Asymptotes

    Example 4

    Consider the rational function

    \[f(x) = \dfrac{(3x^2 + x - 1)}{(x^2 - x - 2)}.\]

    For the numerator, the term \(3x^2\) dominates when \(x\) is large, while for the denominator, the term \(x^2\) dominates when \(x\) is large. Hence as

    \[x \rightarrow \infty,\]

    \[f(x)\rightarrow\dfrac{3x^2}{x^2}=3.\]

    3 is called the horizontal asymptote and we have the the left and right behavior of the graph is a horizontal line \(y = 3\).

    4. Oblique Asymptotes

    Consider the function

    \[f(x) = \dfrac{(x^2 - 3x - 4)}{(x + 3)}\]

    \(f(x)\) does not have a horizontal asymptote, since

    \[\dfrac{x^2}{x}= x \]

    is not a constant, but we see (on the calculator) that the left and right behavior of the curve is like a line. Our goal is to find the equation of this line.

    We use synthetic division to see that

    \[\dfrac{(x^2 - 3x - 4)}{(x + 3)} = x - 6 + \dfrac{14}{(x+3)}.\]

    For very large \(x\),

    \[\dfrac{14}{x} + 3\]

    is very small, hence \(f(x)\) is approximately equal to

    \[x - 6\]

    on the far left and far right of the graph. We call this line an Oblique Asymptote.

    To graph, we see that there is a vertical asymptote at

    \[x = -3\]

    with behavior:

    left down and right up

    The graph has x-intercepts at 4 and -1, and a y intercept at \(-\frac{4}{3}\).

    Exercise

    Graph

    \[\dfrac{(x^3 + 8)}{(x^2 - 3x - 4)}\]

    5. Rational Functions With Common Factors

    Consider the graph of

    \[y = \dfrac{x-1}{x-1}\]

    What is wrong with the picture? When

    \[f(x) = \dfrac{g(x)(x - r)}{h(x)(x - r)}\]

    with neither \(g(r)\) nor \(h(r)\) zero, the graph will have a hole at \(x = r\). We call this hole a removable discontinuity.

    Example

    Graph

    \[\begin{align} f(x) &= \dfrac{(x^2 - 2)}{(x^2 - x - 2)} = \dfrac{(x - 2)(x + 2)}{(x - 2)(x + 1)}.\end{align}\]

    This graph will have a vertical asymptote at \(x =-1\) and a hole at \((2,2)\).

    We end our discussion with a list of steps for graphing rational functions.

    Steps in graphing rational functions:

    • Step 1 Plug in \(x = 0\) to find the y-intercept
    • Step 2 Factor the numerator and denominator. Cancel any common factors remember to put in the appropriate holes if necessary.
    • Step 3 Set the numerator = 0 to find the x-intercepts
    • Step 4 Set the denominator = 0 to find the vertical asymptotes. Then plug in nearby values to fine the left and right behavior of the vertical asymptotes.
    • Step 5 If the degree of the numerator = degree of the denominator then the graph has a horizontal asymptote. To determine the value of the horizontal asymptote, divide the term highest power of the numerator by the term of highest power of the denominator.
    • If the degree of the numerator = degree of the denominator + 1, then use polynomial or synthetic division to determine the equation of the oblique asymptote.
    • Step 6 Graph it!

    Larry Green (Lake Tahoe Community College)

    • Integrated by Justin Marshall.