Basic Rational Functions, Power Functions, and Asymptotes

Homework Schedule and Course Overview

• Homework Assignments:
  • Assignments are provided whenever a course section is completed.
  • One assignment is due tonight.
  • A shorter assignment is assigned tonight and is due on Monday.
  • An additional homework on power functions will be posted, which will be shorter than the toolbox functions assignment.
• Main Topics Covered:
  • Basic rational functions and power functions.
  • Graphing rational and power functions.
  • Asymptotes (horizontal and vertical).
  • Formal description of end behavior.
  • Graphing transformed versions of functions.
  • Domains and symmetric properties.

Basic Rational Functions

• Definition of a Rational Function:
  • A rational function $b(x)$ is defined as a ratio of two polynomials:                  $b(x) = \frac{n(x)}{d(x)}$                  where both $n(x)$ and $d(x)$ are polynomials.
• Domain Conditions:
  • Domain restrictions occur exclusively when the denominator is equal to zero ($d(x) = 0$).
  • The domain of $b(x)$ includes all real numbers except the zeros of the denominator polynomial $d(x)$.
  • By the fundamental properties of polynomials, a polynomial $d(x)$ of degree $n$ has at most $n$ zeros. Consequently, there are at most $n$ domain restrictions in $b(x)$.
• The Reciprocal Function:
  • The simplest basic rational function is the reciprocal function:                  $y = \frac{1}{x}$
  • The function takes any non-zero input $x$ and outputs its multiplicative inverse (reciprocal).
  • Domain: All real numbers except $x = 0$ ($(-\infty, 0) \cup (0, \infty)$). The value $x = 0$ is the root of the polynomial denominator $y = x$.
  • Numerical Behavior for Large Inputs:
    • As the magnitude of the divisor increases, the quotient decreases.
    • Example: $\frac{12}{3} = 4$, whereas $\frac{12}{6} = 2$.
    • As $x$ becomes extremely large in the positive direction ($x \to \infty$), $y$ gets closer to $0$ while remaining positive.
    • As $x$ becomes extremely large in the negative direction ($x \to -\infty$), $y$ gets closer to $0$ while remaining negative.
  • Numerical Behavior for Small Inputs (Near Zero):
    • Dividing by smaller numbers yields larger magnitude outputs.
    • Example: $\frac{24}{2} = 12$, whereas $\frac{24}{8} = 3$.
    • As $x$ gets closer to $0$ from the right ($x \to 0^+$), $y$ becomes an arbitrarily large positive number.
    • As $x$ gets closer to $0$ from the left ($x \to 0^-$), $y$ becomes an arbitrarily large negative number.
• Key Coordinates and Graphing Characteristics:
  • The relation passes the vertical line test and is a valid function.
  • There is a break/gap in the graph at $x = 0$ because the function is undefined at this point.
  • Special Invariant Points:
    • $(1, 1)$ because $\frac{1}{1} = 1$.
    • $(-1, -1)$ because $\frac{1}{-1} = -1$.
  • Odd Symmetry:
    • The function exhibits odd symmetry (symmetry across the origin).
    • Algebraic proof: $f(-x) = \frac{1}{-x} = -\frac{1}{x} = -f(x)$.
    • Geometric reflection across the origin maps the point $(1, 1)$ directly to $(-1, -1)$.

End Behavior and Asymptotes

• Formal End Behavior Notation:
  • As $x \to \infty$, $y \to 0$.
  • As $x \to -\infty$, $y \to 0$.
  • The equation $\frac{1}{x} = 0$ has no solution; the graph approaches $0$ but never reaches or crosses the x-axis.
  • Directional approaches near vertical breaks use directional superscripts:
    • Approaching from the right (larger values): As $x \to 0^+$, $y \to \infty$.
    • Approaching from the left (smaller values): As $x \to 0^-$, $y \to -\infty$.
• Horizontal Asymptotes:
  • Definition: A horizontal line $y = k$ is a horizontal asymptote for a function $v(x)$ if, as $x$ increases or decreases without bound, $v(x)$ approaches $k$:                  $\text{As } x \to \infty, v(x) \to k \quad \text{or} \quad \text{As } x \to -\infty, v(x) \to k$
  • For both $y = \frac{1}{x}$ and $y = \frac{1}{x^2}$, the horizontal asymptote is the line $y = 0$ (the x-axis).
  • Transformed Examples:
    • $y = \frac{1}{x} + 1$: Shifted upward by $1$ unit. As $x \to \pm\infty$, $\frac{1}{x} \to 0$, leaving $y \to 1$. Horizontal asymptote is $y = 1$. The point $(1,1)$ moves to $(1,2)$ and $(-1,-1)$ moves to $(-1,0)$.
    • $y = \frac{1}{x^2} - 2$: Shifted downward by $2$ units. As $x \to \pm\infty$, $\frac{1}{x^2} \to 0$, leaving $y \to -2$. Horizontal asymptote is $y = -2$.
• Vertical Asymptotes:
  • Definition: A vertical line $x = h$ is a vertical asymptote for a function $g(x)$ if, as $x$ approaches $h$ from either side, the absolute value of the function increases without bound:                  $\text{As } x \to h^+ \text{ or } x \to h^-, |g(x)| \to \infty$
  • For $y = \frac{1}{x}$ and $y = \frac{1}{x^2}$, the vertical asymptote is the line $x = 0$ (the y-axis).

The Reciprocal Square Function

• Definition: The reciprocal square function is given by:          $y = \frac{1}{x^2}$
• Domain and Range:
  • Domain: All real numbers except $x = 0$ ($(-\infty, 0) \cup (0, \infty)$).
  • Range: All positive real numbers ($(0, \infty)$). Output values can never be zero or negative because the square of any real number in the domain is strictly positive.
• Symmetry:
  • Exhibits even symmetry (symmetric across the y-axis).
  • Algebraic proof: $f(-x) = \frac{1}{(-x)^2} = \frac{1}{x^2} = f(x)$.
• Rate of Change Properties:
  • Squaring inputs less than $1$ ($|x| < 1$) results in smaller denominator values, causing the output to blow up faster toward positive infinity than in $\frac{1}{x}$.
  • Squaring inputs greater than $1$ ($|x| > 1$) produces larger denominator values, causing output values to decay toward zero faster than in $\frac{1}{x}$.

Transformations of Basic Rational Functions

• General Transformation Equations:
  • Reciprocal Function Form:                  $f(x) = \frac{a}{x - h} + k$
  • Reciprocal Squared Function Form:                  $f(x) = \frac{a}{(x - h)^2} + k$
• Role of Transformation Parameters:
  • $a$: Controls vertical stretching, vertical compression, and vertical reflection (flipping across the x-axis if $a < 0$).
  • $h$: Determines the horizontal shift. Shifts the vertical asymptote to $x = h$.
  • $k$: Determines the vertical shift. Shifts the horizontal asymptote to y = k$.\n* **Example 1: Graphing an Equation**:\n * Given function: f(x) = \frac{1}{x - 2} + 1\n * **Parameters**: a = 1$(no scaling or reflection),$h = 2$,$k = 1\n * **Asymptotes**:\n * Vertical asymptote at x = 2\n * Horizontal asymptote at y = 1\n * **Intercepts**:\n * y$**-intercept**: Set$x = 0:\n            \n            f(0) = \frac{1}{0 - 2} + 1 = -\frac{1}{2} + 1 = \frac{1}{2}\n            \n            Point: (0, \frac{1}{2}).\n * x$**-intercept**: Set$y = 0:\n            \n            0 = \frac{1}{x - 2} + 1 \implies -1 = \frac{1}{x - 2} \implies -(x - 2) = 1 \implies -x + 2 = 1 \implies x = 4\n            \n            (Note: Solving algebraically yields x = 1$, corresponding to$(1, 0)).\n* **Example 2: Determining an Equation from a Graph**:\n * Graph shape is symmetric about its vertical asymptote with both branches going in the same direction, matching \frac{1}{x^2}$with$a = 1.\n * Vertical asymptote is located at x = 2$($h = 2).\n * Horizontal asymptote is located at y = -1$($k = -1).\n * **Resulting Equation**:\n        \n        f(x) = \frac{1}{(x - 2)^2} - 1\n        \n\n# Power Functions and Root Functions\n\n* **Connection to Negative Exponents**:\n * \frac{1}{x} = x^{-1}\n * \frac{1}{x^2} = x^{-2}\n* **Definition of Power Functions**:\n * For any real number p$and variable$x$, functions of the form$f(x) = x^p are called power functions.\n* **Definition of Root Functions**:\n * When p = \frac{1}{n}$for an integer$n \ge 2, power functions are specifically termed root functions:\n        \n        x^{\frac{1}{n}} = \sqrt[n]{x}\n        \n* **Comparison of Growth Rates**:\n * For different exponent values, function growth rates order as follows (for large x):\n        \n        x^{-4} < x^{\frac{2}{3}} < x^1 < x^{\frac{3}{2}} < x^2\n        \n * Larger exponent magnitudes |p| result in faster function growth rates.\n* **Domain Rules for Power Functions** f(x) = x^p$(where$p = \frac{m}{n} > 0):\n * **Odd Root Denominator (n$is odd)**: Domain is all real numbers ($(-\infty, \infty)). Odd roots (e.g., cube root) can evaluate both negative and positive numbers.\n * **Even Root Denominator (n$is even)**: Domain is strictly non-negative real numbers ($[0, \infty)). Even roots cannot evaluate negative numbers.\n * **Irrational Exponent (p$cannot be written as a fraction)**: Domain is non-negative real numbers ($[0, \infty)$), such as$x^{\frac{1}{\pi}}.\n * The numerator m$does not restrict domain behavior because integer powers (e.g.,$x^2, x^3, x^4) are defined for all real numbers.\n* **General Transformation Form for Power Functions**:\n    \n    f(x) = a(x - h)^p + k\n    \n * a: Vertical scaling (stretch/compression) and vertical reflection.\n * h$: Horizontal translation (subtracting$h$directly from$x).\n * k$: Vertical translation (adding$k$$ to the output).

Questions and Classroom Discussion

• Absence Notification:
  • Student Prompt: Students notified the instructor that they will be absent from class due to traveling to Minneapolis to attend a concert early that day.
  • Concert Details: The concert features the parody band Star Bomb (associated with Dan and Aaron, who also perform in NSP / Ninja Sex Party).
  • Instructor Response: Attendance is not formally taken, and no in-class group activities are missed. Course notes and homework assignments will be published online. Missing work must be completed by the specified due date.