Functions, Domain and Range

Domain and Range of a Relation

  • The given relation is a set of ordered pairs, written with braces: { (4, 9), (-4, 9), (2, 3), (10, -5) } (the transcript shows pairs like 4,9; -4,9; 2,3; 10,-5).

  • Domain: the set of x-values used by the relation.

    • From the pairs: x-values are { 4, -4, 2, 10 }.

    • In set notation: x∈R∣−?≤x≤?{ x \in \mathbb{R} \mid -? \le x \le ? } is not appropriate here because we have a finite set; more directly, the domain is the set of these x-values: 4,−4,2,10.{4, -4, 2, 10}.

  • Range: the set of y-values produced by the relation.

    • From the pairs: y-values are { 9, 9, 3, -5 } which, as a set, is −5,3,9{ -5, 3, 9 } (note that 9 is used twice, but sets only list distinct values).

    • If ordered by appearance, you might see 9, 9, 3, -5, but the range as a set uses distinct values. In sorted order: −5,3,9.{-5, 3, 9}.

  • Endpoints and notation:

    • For a finite set of points, we don’t use an interval; we list the sets of x and y values as above.

    • If a domain or range were to be described as an interval, you’d use bracket notation when endpoints are included (closed) and parentheses when endpoints are not included (open). The endpoints being included is indicated by closed circles in graphs; otherwise use open circles.

  • Is this relation a function?

    • A function requires every input x to map to exactly one output y.

    • Mapping from the domain to the range:

    • 4 → 9

    • -4 → 9

    • 2 → 3

    • 10 → -5

    • Each input has exactly one output, so this is a function, even though two inputs map to the same output (4 and -4 both map to 9).

    • If we added (10, 3) (i.e., 10 maps to two outputs: 3 and -5), it would no longer be a function because the input 10 would have multiple outputs.

  • The vertical line test (graphical test for functions)

    • If a graph has a vertical line that intersects the graph at more than one point, the relation is not a function.

    • Intuition: a single input x would yield multiple y-values.

    • If a vertical line intersects at most one point for all x, the relation is a function.

  • Lines and function status

    • Horizontal lines are functions (a horizontal line y = c passes the vertical line test for all x).

    • Any line that is not vertical (i.e., a line with slope, like y = mx + b) is also a function.

    • A vertical line (x = c) is not a function because it assigns multiple y-values to the single input x = c.

  • Function notation and terminology

    • A function expresses output as a dependent variable of an input: y is a function of x.

    • The name of the function is a symbol like f or g (lowercase for f, sometimes Greek letters).

    • Notation: if the function is named f, then f(x) denotes the output when the input is x.

    • The input value is the independent variable inside the parentheses; the output is the dependent variable (the value of f(x)).

- If a linear function is given by y = mx + b, it is a function for all real x (except vertical lines, which are not of the form y = mx + b).

-# Function notation and examples
-Function notation lets us talk about outputs without listing all ordered pairs.
-Example: Given y = -3x + 2, this is a linear function (a line with slope -3 and y-intercept 2).
-Since it’s a function, we can rewrite it using function notation:
-f(x)=−3x+2.f(x) = -3x + 2.
-Then, to evaluate, substitute any x value into the rule: for example,\
-f(0)=−3⋅0+2=2.f(0) = -3\cdot 0 + 2 = 2.\
-So the ordered pair is \(0, 2\).

-If we wanted to know f(a) for a general a, we substitute a for x: f(a) = -3a + 2.

-# Example: A quadratic function and domain/range
-Let g(x) = x^2 - 2x. We’re asked to find g(-3) and to determine the domain and range (for a specified domain if given).
-Compute g(-3):
-g(−3)=(−3)2−2(−3)=9+6=15.g(-3) = (-3)^2 - 2(-3) = 9 + 6 = 15. \
-Thus the ordered pair corresponding to x = -3 is (-3, 15).
-Determine a domain and range based on the given inputs/outputs.
-If the domain is the x-values used by the relation from the context (here, the course notes show x-values from -3 to 4), then:
-Set notation domain: x∈R ∣ −3≤x≤4.{ x \in \mathbb{R} \,|\, -3 \le x \le 4 }.\
-Interval notation domain: $$[-3, 4