Section 1.3 - 1.4: Information from Graphs and Combining Functions

Local Maxima, Local Minima, and Global Extremes

• Definition of Local Maximum: The largest function value in a given "neighborhood" or specific area of the graph.
• Definition of Local Minimum: The smallest function value in a given neighborhood of the graph.
• Definition of Global (Absolute) Extremes:
  • Global Maximum: The single highest point on the entire graph. Not every function possesses a global maximum.
  • Global Minimum: The single smallest point on the entire graph. Not every function possesses a global minimum.
• Standard Notation for Extremes:
  • Findings must be presented as: "[Function Value] at $x = [Coordinate]$".
  • Example: $8$ at $x = -2$.
  • Warning: Using point format (e.g., $(-2, 8)$) for maxima and minima results in a one-point deduction on assessments.
• Reachability and Infinity:
  • For a value to be a maximum or minimum, the function must actually reach that value.
  • Infinity (\n\infty) can never be a maximum or minimum because it is unreachable; one can always choose a number closer to infinity than the previous choice.
• Supremum and Infimum:
  • Supremum: A point that a function gets infinitely close to but never touches, which would otherwise be a maximum.
  • Infimum: A point that a function gets infinitely close to but never touches, which would otherwise be a minimum.

Graph Analysis: Intervals of Increase and Decrease

• Interval Type: Always use x-intervals for increasing and decreasing sections.
  • The focus is on where the behavior is happening, rather than the function value itself.
• Notation Rules: Always use parentheses for intervals of increasing and decreasing.
  • Reason: At the exact turnaround points (the endpoints of these intervals), the function is neither increasing nor decreasing; it is simply "being" or turning.
  • Each piece of an interval (the start and the end cap) is worth half a point on grading.
• Identifying Intervals via Graphing:
  • Moving left to right, if the function moves "north," it is increasing.
  • Moving left to right, if the function moves "south," it is decreasing.
• Relation to Domain:
  • The union of the intervals of increasing and decreasing, along with the turnaround points, should typically equal the domain of the function.

Analyzing Specific Function Examples

• Piecewise Function Example:
  • Branch 1: $f(x) = -x^3$ where $x < 0$.
  • Branch 2: $f(x) = 2x - 1$ where $x \ge 0$.
  • Local Max: $8$ at $x = -2$.
  • Local Min: $-1$ at $x = 0$. Note that even though the first branch approaches $(0,0)$, it does not reach it, whereas the second branch reaches $(0,-1)$.
  • Increasing: $(0, \infty)$.
  • Decreasing: $(-2, 0)$.
• Exponential Function Example ($f(x) = 2^x$):
  • Local Max/Min: Does not exist (DNE). The function approaches \n\infty and $0$ but never reaches either.
  • Interval of Increase: Negative infinity to positive infinity ($(-\infty, \infty)$).
  • Interval of Decrease: Does not exist. This is an "always increasing" function.
  • Negative Exponents: A negative exponent (e.g., $2^{-800} = \frac{1}{2^{800}}$) results in a very small positive number, not a negative number.
• The Cubic Function Example ($f(x) = x^3$):
  • Saddle Point / Inflection Point: At $x = 0$, the graph flattens out but does not change direction. It increases until it hits zero and continues to increase afterward.
  • Extremes: No local maxima or minima exist.

Combining Functions: Arithmetic Operations

• Definitions: Let $f(x)$ and $g(x)$ be functions.
  • Addition: $(f + g)(x) = f(x) + g(x)$.
  • Subtraction: $(f - g)(x) = f(x) - g(x)$.
  • Multiplication: $(fg)(x) = f(x) \times g(x)$.
  • Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, provided $g(x) \not\equiv 0$.
• Evaluation Principle: You can either combine the formulas first and then plug in the number, or plug the number into each function individually and then perform the operation—both methods yield the same result.
• Domain of Combined Functions: The domain is the intersection of the domains of the individual functions.
  • Intersection: The value must be valid for both functions simultaneously. If a value is undefined for one piece, the combined function is undefined at that value.
  • Example: If $f(x) = x^2$ (domain: all real numbers) and $g(x) = \frac{1}{x}$ (domain: all reals except $0$), the combination's domain is all real numbers except zero.

Function Composition

• Notation: $(f \circ g)(x)$, read as "$f$ composed with $g$ of $x$."
• Definition: $(f \circ g)(x) = f(g(x))$. The inner function $g(x)$ is substituted into the outer function $f(x)$.
• Calculation Order: Always work from the inside out.
  1. Evaluate the inner function for a value ($g(a)$).
  2. Use that result as the input for the outer function ($f(g(a))$).
• Commutativity Warning: Function composition is not commutative. In general, $(f \circ g)(x) \neq (g \circ f)(x)$.
  • Example with $f(x) = -3x + 2$ and $g(x) = 2x^2 - 7$ at $x = 5$:
    • $f(g(5)) = -127$.
    • $g(f(5)) = 331$.
• Self-Composition (Identity): A function can be composed with itself, such as $(f \circ f)(x)$. If $(f \circ f)(x) = x$, the function is its own inverse.

Advanced Polynomial and Rational Composition

• Polynomial Multiplication Stacking Method: Instead of horizontal distribution (side-by-side), polynomials can be multiplied by stacking them vertically, similar to multi-digit integer multiplication.
  • This method helps line up like terms ($x^4, x^3, x^2$, etc.) automatically.
  • Example: $(2x^2 + 3x + 1) \times (2x^2 + 3x + 1)$.
  • This avoids losing terms during the distribution process.
• Rational Composition via "The Tuxedo" Method: When composing rational functions (fractions within fractions), you can simplify the complex fraction by multiplying the entire numerator and denominator by the Least Common Denominator (LCD).
  • This "changes the look" (like wearing a tuxedo vs. jeans) without changing the identity of the expression.
  • Example with $f(x) = \frac{2x + 1}{3x - 2}$, calculating $(f \circ f)(x)$:
    • Substitute the whole fraction into itself.
    • Multiply the top and bottom by $(3x - 2)$.
    • This clears the internal denominators, simplifying the complex fraction to $x$.

Questions & Discussion

• Q: Does it matter if you use brackets or parentheses for increasing/decreasing?
  • A: Yes. Always use parentheses. At the peak or valley, the function is momentarily steady and not doing either action.
• Q: Are we going over concave up and down?
  • A: No. Concavity is relegated to Calculus 1.
• Q: Will the difference quotient be on the test?
  • A: Yes, but the formula will be provided.
• Q: What is the rule for common denominators in complex fractions?
  • A: Multiply by the common denominator over itself (essentially multiplying by $1$) to clear fractions quickly.

Analogies and Illustrative Examples

• Howie Mandel Analogy: Dealing with points the function approaches but does not reach (like an open circle). You can get as close as a handshake, but you'll only ever get a "fist bump" (germaphobe reference).
• Timothy Leary: Referenced jokingly in regards to a "trippy" looking piecewise graph.
• Apollo 13 Oxygen Levels: Used to illustrate $-x^3$ behavior; oxygen levels drop, level out (inflection point), then continue to drop.
• Turducken/Pie-caken: Used to explain function composition—shoving one thing (a chicken or a pie) inside another (a turkey or a cake).
• Intersection Analogy: Being at the intersection of 152nd and Meridian means you are on both streets simultaneously, just as a domain intersection requires validity in both functions.