MT

Reading the Draft: Functions (Introduction)

Reading the Draft: Functions

  • Definition of a function (very loosely): a dependence of one quantity on another quantity. In notation, a function is often written as f, and when we say f(a) we mean the output value when the input is a.
  • Function idea in plain terms: an output quantity that depends on an input quantity.
  • Independent vs dependent variable:
    • Independent variable (the input we choose) is the variable you pick.
    • The output (dependent variable) depends on what input you chose.
    • Mnemonic: independent starts with I (I for input you can change).
  • Key requirement for a function:
    • Input values cannot have more than one output value. In other words, for each input, there is a unique output.
    • It’s okay for different inputs to map to the same output (many inputs can share the same output).
  • Real-life examples of functions:
    • Temperature of coffee as a function of time since you poured it.
    • Temperature depends on how much time has passed.
    • Distance traveled as a function of speed while driving a car.
    • Distance is a function of speed (and time), given starting point.
    • Cost of mailing a package as a function of its weight (and possibly other factors).
    • Weight is a primary input affecting cost, though other factors can also play a role.
  • A concrete example function: f(x) = x^2 + 4
    • This is a common way functions are written: the input is x, the output is f(x).
    • Evaluating: to find f(3), substitute 3 for x: f(3) = 3^2 + 4 = 13.
    • Evaluating with a negative input: f(-2) = (-2)^2 + 4 = 4 + 4 = 8.
    • Important caution about parentheses:
    • You must square the quantity you plug in. If you write f(-2) = -2^2 + 4, that would be incorrect because exponentiation binds tighter than the unary minus. The correct interpretation uses parentheses: (-2)^2 = 4.
  • Domain vs. real numbers:
    • The domain of this function f(x) = x^2 + 4 is all real numbers: ext{Dom}(f) = (-\infty, \infty).
    • In interval notation, infinity is written as (-
      ofty, ofty) (usually written as (-
      leftarrow o) in plain text).
  • Graphical perspective: inputs are on the horizontal axis (the x-axis).
    • A graph represents a function if it passes the vertical line test: any vertical line intersects the graph at most once.
    • If a vertical line intersects the graph more than once, the graph is not a function (e.g., a circle).
    • Quick mental check: if you can draw a vertical line that hits the graph in two or more points, it’s not a function.
  • Worked example with a graph (domain from a given graph):
    • To determine the domain from a graph, examine where x-values have corresponding y-values.
    • If for every real number x there is a corresponding y, the domain is (-
      inf,
      inf) (all real numbers).
    • If there are gaps, express the domain as a union of intervals, e.g., (-
      inf, a) \,\cup \,(b,
      inf), where the gap is between a and b.
    • In the instructor’s example: left of 1 there are some x-values with y-values, right of 2 every x-value has a y-value, suggesting the domain is all real numbers; if you had a break (e.g., a hole at 0), you’d write (-
      inf, 0) \,\cup \,(0,
      inf).
  • Range from a graph:
    • Instead of asking for input values (x-values), you can also discuss which y-values are actually used.
    • Example notes from the session: there were no negative y-values visible on the particular graph, meaning the range did not include negative numbers.
    • The point (0,0) being on the graph would indicate that 0 is in the range if the graph passes through the origin.
  • Analyzing f(x) values geometrically:
    • Values where f(x) = 0 correspond to the x-intercepts of the graph.
    • Values where f(x) > 0 correspond to y-values above the x-axis; if no portion of the graph lies below the x-axis, then there are no x-values with f(x) < 0.
    • Values where f(x) < 0 would occur only if the graph has points below the x-axis.
  • Practice question style (abstract): finding the set of x-values for which f(x) = g(x) (intersection with another function g):
    • When solving f(x) = g(x), you are looking for x-values where the two graphs share the same y-value.
    • On a graph, these are the intersection points of the two curves.
    • Example scenario from the lecture: there are several intersection points (three were observed) where the two functions share the same y-value.
    • Practical way to locate intersections on a graph: compare which function is higher at a given x to decide which x-values satisfy f(x) = g(x), or draw a horizontal line at the target y-value and see where it intersects both graphs.
  • A concrete note on a 25-degree example mentioned in the session:
    • To find where the combined functions equal a horizontal line, draw a horizontal line at that y-value (e.g., y = 25).
    • Look at the x-values where this horizontal line intersects the graphs of the functions; those x-values are the solutions to f(x) = 25 (or f(x) = g(x) when comparing two graphs).
    • In the instructor’s discussion, there were two intersection x-values for the line y = 25 with the graphs, illustrating how one reads off solutions from the graph.

Interval notation and union for domains

  • When a domain excludes a single point (e.g., exclude 0):
    • The domain can be written as (-
      inf, 0) \cup (0,
      inf).
    • In many learning platforms (e.g., WebAssign), you can also omit the explicit union symbol and place the intervals next to each other if allowed by the system (e.g., two separate interval entries).
  • Important reminder about infinity:
    • Infinity is not a number, so statements like "add one to infinity" are informal intuition rather than arithmetic truth.
    • The rule examples given (e.g., frac{2}{2} = 1, frac{3}{3} = 1, frac{4}{4} = 1) illustrate that real-number rules hold, but when talking about limits or infinity, one must be careful: frac{ ext{infinity}}{ ext{infinity}} is not defined as a real number and is not equal to 1.
  • Quick takeaway: use interval notation to describe domains that are not all real numbers, and remember to use the union symbol to combine disjoint intervals when appropriate.

Connections to broader concepts

  • Foundational ideas:
    • A function assigns exactly one output to each input within its domain.
    • The domain and range describe the complete set of inputs and outputs that actually occur for a function.
  • In real-world modeling:
    • Choosing the independent variable (the input) reflects what you can control or vary.
    • The dependent variable represents the outcome you observe or predict based on the input.
  • Practical implications:
    • Misinterpreting input/output or misplacing parentheses can lead to incorrect results (e.g., confusing x^2 with (-x)^2).
    • When reading graphs, the vertical line test is a quick diagnostic for whether a relation is a function.
    • Infinity and limits require careful mathematical treatment beyond simple arithmetic intuition.

Summary of key formulas and ideas (quick reference)

  • Function notation and evaluation: f(x) = \text{expression in } x, f(a) = \text{output when input } a.
  • Example evaluation: f(3) = 3^2 + 4 = 13, f(-2) = (-2)^2 + 4 = 8.
  • Domain of f(x) = x^2 + 4: ext{Dom}(f) = (-\infty, \infty).
  • Range for this example: f(x) \ge 4 \Rightarrow \text{Range}(f) = [4, \infty).
  • Vertical line test: a graph is a function iff every vertical line intersects it at most once.
  • Domain with a hole at 0: (-\infty, 0) \cup (0, \infty).
  • Intersections of two functions: solve f(x) = g(x); x-values where the graphs cross.
  • Reading a horizontal line to locate y-values: for a given y, identify x-values where the graph attains that y (intersection points with the line y = constant).
  • Important caution about infinity: treat ∞ as a concept in limits, not a real number; arithmetic rules differ from finite real numbers.

Note: key ideas to remember for exams

  • A function assigns exactly one output for each input, with input domain as the allowed x-values.
  • The independent variable is what you choose; the dependent variable is the outcome/output.
  • Use the vertical line test to verify a graph is a function.
  • When evaluating functions, mind parentheses to ensure correct operations (e.g., (-2)^2 not -2^2).
  • For graph-based domain/range, read off x-values that yield y-values and y-values that are actually attained.
  • Intersection with another function is found at points where the two graphs share the same y-value for the same x-value (i.e., where f(x) = g(x)).
  • Use interval notation and unions to describe domains with holes or multiple disjoint pieces.$$