Identifying Functions Using Graphs page 20

Graphical Representation of Functions

• Functions can be visually represented using graphs, which display numerous input-output pairs in a small, organized space.

• The visual presentation provided by a graph makes mathematical relationships significantly easier to comprehend and analyze.

• Axes Convention:

  • Input values are plotted along the horizontal axis (the $x$-axis).

  • Output values are plotted along the vertical axis (the $y$-axis).

• Variable Naming and Functional Notation:

  • The standard naming convention designates the input value as $x$ and the output value as $y$.

  • When the function is named $f$, $y$ is described as a function of $x$, expressed algebraically as $y = f(x)$.

• Definition of a Function's Graph:

  • The graph of a function $f$ consists of the set of all ordered pairs $(x, y)$ in the coordinate plane that satisfy the relationship $y = f(x)$.

  • If a function is defined for only a finite set of discrete input values, its graph consists solely of isolated points, where the $x$-coordinate of each point is an input value and the $y$-coordinate is the corresponding output value.

  • If a function is defined across a continuous set of inputs, the set of all points $(x, y)$ satisfying $y = f(x)$ forms a continuous curve.

• Specific Point Analysis from Graphical Examples:

  • Black dots marked on a graph explicitly show specific input-output values.

  • A point at $(0, 2)$ indicates that when the input $x = 0$, the output is $y = 2$, which is written in function notation as $f(0) = 2$.

  • A point at $(6, 1)$ indicates that when the input $x = 6$, the output is $y = 1$, which is written in function notation as $f(6) = 1$.

  • The curve shown in the graph passes through the coordinates $(0, 2)$ and $(6, 1)$, proving that both ordered pairs satisfy the equation $y = f(x)$.

The Vertical Line Test

• Definition and Application:

  • The vertical line test is a visual method used to determine whether a given graph represents a function.

  • A vertical line contains all points in the coordinate plane that share one specific input $x$-value.

• Mechanism of Intersections:

  • The $y$-value of any point where a vertical line intersects a graph represents an output value corresponding to that specific input $x$-value.

• Testing Criterion for Functions:

  • If any vertical line can be drawn that intersects a graph more than once, then the graph does not define a function.

  • Multiple intersection points mean that a single input $x$-value maps to more than one output $y$-value.

  • Fundamental Rule of Functions: A relation qualifies as a function if and only if every input value is paired with exactly one output value.