Function concept and notation Knowledge-Based Study

Study notes

1. Definition of a Function

   In mathematics, a relation is any set of ordered pairs connecting inputs to outputs. A function is a special type of relation with a strict rule:

   Every element in the input set (domain) corresponds to exactly one element in the output set (range).

   Key Terminology

   • Domain: The set of all allowable input values, typically denoted by $x$.
   • Codomain: The target set that contains all potential output values.
   • Range: The set of all actual output values produced by the function, which is a subset of the codomain.

   Think of a function as an input-output machine. When you input an object or number, the machine processes it according to a fixed rule and produces one unique result. If an input can yield two or more different outputs, the relation is not a function.

   Understanding Function Notation

   Instead of naming equations with variables such as $y = 2x + 3$, we frequently use function notation:

   $$f(x) = 2x + 3$$

   This notation is read aloud as "f of x equals two x plus three".

   Anatomy of the Notation

   • $f$ is the name of the function. Other common letters include $g$, $h$, or capital letters like $F$.
   • $x$ inside the parentheses represents the input (independent variable).
   • $f(x)$ represents the output value corresponding to $x$ (dependent variable). Therefore, $y = f(x)$.

   Crucial Note: The expression $f(x)$ does not mean $f$ multiplied by $x$. The parentheses indicate substitution into the function's rule.

   How to Evaluate a Function

   To evaluate a function for a specific value, substitute that value everywhere the independent variable appears in the formula.

   Example 1: Numerical Input

   Given $f(x) = x^2 - 4x + 1$, find $f(3)$:

   • Replace $x$ with $3$:
   • $f(3) = (3)^2 - 4(3) + 1$
   • $f(3) = 9 - 12 + 1$
   • $f(3) = -2$

   The ordered pair on the graph of $f$ is $(3, -2)$.

   Example 2: Algebraic Input

   Given $g(t) = 5 - 2t$, find $g(a + 1)$:

   • Replace $t$ with $(a + 1)$:
   • $g(a + 1) = 5 - 2(a + 1)$
   • $g(a + 1) = 5 - 2a - 2$
   • $g(a + 1) = 3 - 2a$

   Identifying Functions

   A relation can be represented using various formats. In each format, we check whether any input has multiple outputs.

   1. Sets of Ordered Pairs

   Consider the relation $R = \{(1, 4), (2, 7), (3, 4)\}$. Each input ($1, 2, 3$) has only one output. Thus, $R$ is a function. Note that different inputs can share the same output (both $1$ and $3$ give $4$).

   Now consider $S = \{(2, 5), (2, 9), (4, 1)\}$. The input $2$ is paired with two distinct outputs ($5$ and $9$). Thus, $S$ is not a function.

   2. Graphical Representation: The Vertical Line Test

   If a relation is graphed on the Cartesian plane, we use the Vertical Line Test:

   A curve in the coordinate plane represents a function of $x$ if and only if no vertical line intersects the curve more than once.

   If a vertical line intersects a graph at two or more points, it means a single $x$-value corresponds to multiple $y$-values, violating the definition of a function.