2.1

Section 2.1 Functions

  • What is a relation and what is a function

    • A relation is a correspondence between two sets. If x and y are elements of these sets and a relation exists between x and y, we say that x corresponds to y (y depends on x), and we write x ⟶ y (or x maps to y).
    • A function is a special kind of relation with a stricter rule: every element of the domain (input set) must map to exactly one element of the range (output set).
    • Example given: a relation where the domain is {410, 430, 540, 580, 600, 750} and the range is {19, 23, 24, 29, 33}. It is acceptable for different domain elements to map to the same range value (one range value can have multiple domain preimages).
    • Not a function example: if an input maps to more than one output (e.g., 0.86 has two prices assigned to it), then the relation is not a function.

  • Important terminology

    • Domain: the set of all first components of the ordered pairs (the inputs).
    • Range: the set of all second components of the ordered pairs (the outputs).
    • For a function f, we write y = f(x) where x is the independent variable (argument) and y is the dependent variable (value of f at x).
    • Notation: x ⟶ y (or x maps to y) represents the relation; f denotes the function, and f(x) denotes the output corresponding to input x.

  • Practice: determine whether the given relations are functions

    • a) {(2, 3), (4, 1), (3, –2), (2, –1)}
    • Domain elements: {2, 4, 3}. The input 2 maps to two outputs (3 and –1).
    • Conclusion: Not a function.
    • b) {(-2, 3), (4, 1), (3, –2), (2, –1)}
    • Domain: { -2, 4, 3, 2 }
    • Range: { 3, 1, –2, –1 }
    • Each domain element maps to exactly one output.
    • Conclusion: Function. Domain = ext{Domain} = igl{-2, 2, 3, 4igr}, Range = ext{Range} = igl{3, 1, -2, -1igrigr
      ightarrow ext{(in any order)}}
    • c) {(4, 3), (3, 3), (4, –3), (2, 1)}
    • Domain contains 4 twice, mapping to two outputs (3 and –3).
    • Conclusion: Not a function.

  • Determine if an equation defines y as a function of x

    • In general, an equation defines y as a function of x if, for every x in the domain, there is exactly one corresponding y.
    • If a given x yields more than one y, the equation does not define y as a function of x.
    • Quick mental checks: solving for y and applying the rule that each x must give a single y is a practical approach. If an equation yields two possible y-values for some x (e.g., squaring both sides, implicit relations), it is not a function of x.
    • Function machine concept (below) reinforces that a function takes a single input and produces a single output.

  • FUNCTION MACHINE and function notation

    • A function can be viewed as a machine: input x goes in, output f(x) comes out.
    • The rule is that y = f(x) gives a single output for each input x.
    • The term f stands for the function; f(x) is the output corresponding to input x.

  • Summary: Important Facts About Functions

    • (a) For each x in the domain of f, there is exactly one image f(x) in the range; however, an element in the range can result from more than one x in the domain.
    • (b) f is the symbol used to denote the function; it is symbolic of the rule or equation that maps x to f(x).
    • (c) If y = f(x), then x is the independent variable (or argument) and y is the dependent variable (the value of f at x).

  • Applications: A rectangular garden problem (area as a function of width)

    • Given: a rectangular garden with a perimeter of 100 feet.
    • Let width = w and length = L.
    • Perimeter constraint: P=2w+2L=100.P = 2w + 2L = 100.
    • Express length in terms of width: L=50−w.L = 50 - w.
    • Area as a function of width: A(w)=wimesL=w(50−w)=50w−w2.A(w) = w imes L = w(50 - w) = 50w - w^2.
    • Domain considerations: both width and length must be nonnegative.
    • From L = 50 - w ≥ 0, we get wextmustsatisfy0 ≤ w≤ 50.w ext{ must satisfy } 0 \,\le\, w \le \,50.
    • Therefore, the function is A(w)=50w−w2A(w) = 50w - w^2 with domain 0≤w≤500 \le w \le 50 (inclusive, allowing degenerate zero-area cases at the endpoints).
    • Note on interpretation: The domain of A as a function of width reflects all feasible widths given a fixed perimeter; the range would be the possible areas produced by those widths.

  • Connections and real-world relevance

    • Functions model real mappings where each input has a single output (e.g., price vs. size, growth models, area as a function of one dimension).
    • The domain and range concepts help understand what inputs are allowable and what outputs can occur, important for modeling and optimization.
    • The “not a function” scenarios underscore the importance of the one-output rule for function quality and for well-defined computations (e.g., when predicting or calculating values).

  • Quick practical tips

    • To test if a set of ordered pairs is a function, check for any repeated first components with different second components.
    • If you have an equation and want to know if it defines y as a function of x, try solving for y and ensure a unique y for each x in the domain.
    • When expressing relations as functions, explicitly state the domain and range if possible, and use the notation y = f(x) to emphasize the functional relationship.

  • Key equations to remember

    • Function rule form: y=f(x)y = f(x)
    • Domain and range notation (example): extDomain=410,430,540,580,600,750ext{Domain} = {410, 430, 540, 580, 600, 750}, Range=19,23,24,29,33\text{Range} = {19, 23, 24, 29, 33}
    • Function area example: A(w)=w(50−w)=50w−w2, 0≤w≤50.A(w) = w(50 - w) = 50w - w^2,\, 0 \le w \le 50.