Basics of Functions and Their Graphs

Basics of Functions and Their Graphs

Definitions: Relation, Domain, and Range

A relation is any set of ordered pairs. The domain of a relation is the set of all first components (inputs), and the range is the set of all second components (outputs).

• Domain: All possible input values (usually x-values).

• Range: All possible output values (usually y-values).

• Relations can be represented in various forms: tables, graphs, mappings, or sets of points.

Example Representations:

• Table: Shows pairs of input and output values.

• Graph: Plots the ordered pairs on a coordinate plane.

• Mapping: Illustrates how each input is paired with an output.

• Set of Points: Example: {(0, 187), (1, 212), (2, 225), (3, 238)}

Definition: Function

A function is a special type of relation in which each element of the domain corresponds to exactly one element in the range. In other words, for every input, there is only one output.

• Function Notation: $f(x)$ denotes the value of the function at $x$.

• Example: The set {(1, 2), (3, 4), (6, 5), (8, 5)} is a function because each input has only one output.

• Non-Example: The equation $x^2 + y^2 = 1$ does not represent $y$ as a function of $x$ because some $x$-values correspond to two $y$-values.

Evaluating a Function

To evaluate a function, substitute the given input value into the function's formula.

• Example: Given $f(x) = x^2 - 2x + 7$, evaluate $f(-5)$:

$\begin{align*} f(-5) &= (-5)^2 - 2(-5) + 7 \\ &= 25 + 10 + 7 \\ &= 42 \end{align*}$

Graphing Functions by Plotting Points

To graph a function, select several input values, compute the corresponding outputs, and plot the resulting points on the coordinate plane. Connect the points if the function is continuous.

• Example: Graph $f(x) = 2x$ and $g(x) = 2x - 3$ for $x = -2, -1, 0, 1, 2$. The graph of $g$ is the graph of $f$ shifted downward by 3 units.

Equation vs Expression

• Equation: A mathematical statement where two expressions are set equal to each other (e.g., $3x + 2 = y$).

• Expression: A mathematical phrase that does not include an equality sign (e.g., $3x + 2$).

Vertical Line Test for Functions

The vertical line test is used to determine if a graph represents a function. If any vertical line intersects the graph more than once, the graph does not represent a function.

Obtaining Information from a Function's Graph

You can extract valuable information from a function's graph, such as the value of the function at a specific input, or for which inputs the function attains a certain output.

• Example: If $f(5) = 400$, then at $x = 5$, the output is 400.

• To find $x$ such that $f(x) = 100$, look for $x$-values where the graph reaches a height of 100.

Identifying Domain and Range from a Function’s Graph

To find the domain from a graph, look for all $x$-values (inputs) that correspond to points on the graph. To find the range, look for all $y$-values (outputs) that correspond to points on the graph.

• Interval Notation: Used to describe continuous sets of numbers, e.g., $[a, b]$ (closed interval), $(a, b)$ (open interval).

• Set Notation: Describes the set of all $x$ such that $a \leq x \leq b$.

Identifying Intercepts from a Function’s Graph

The x-intercepts are the points where the graph crosses the $x$-axis (i.e., where $y = 0$). The y-intercept is the point where the graph crosses the $y$-axis (i.e., where $x = 0$). A function can have multiple $x$-intercepts but at most one $y$-intercept.

• Example: For a function $f(x)$, if the graph crosses the $x$-axis at $x = -1, 2, 3$ and the $y$-axis at $y = -6$, then the $x$-intercepts are $x = -1, 2, 3$ and the $y$-intercept is $y = -6$.

Summary Table: Representations of a Relation

Representation	Description	Example
Table	Lists input-output pairs	See Table 1 above
Graph	Plots points on coordinate axes	See Figure 2 above
Mapping	Shows correspondence between sets	See Figure 3 above
Set of Points	Explicitly lists ordered pairs	{(0, 20), (1, 19.2), ...}