Identifying Functions, Function Notation, and Graph Analysis

A relation is a set of ordered pairs. To determine if a relation is a function visually, a vertical line test is employed.
Vertical Line Test (VLT) Definition: A graph on the coordinate plane defines $y$ as a function of $x$ if no vertical line intersects the graph in more than one point.
Example Analysis (Relation A): In a specific relation, the point $(1, 1)$ may be the only coordinate where the $x$ -value is assigned exactly one $y$ -value. If a vertical line is drawn at other $x$ -values, such as $x = -2$ , and it intersects the graph at multiple points, the relation fails the vertical line test and is not a function.
Example Analysis (Relation B): A graph consisting of plotted points is not a function if any vertical line passes through two or more points. For instance, if vertical lines intersect the graph in two places at $x = 2$ , it fails to be a function at that specific location.
Negated Condition: If a vertical line can be drawn that intersects a graph in more than one place, then the graph is not a function.

Function notation is the standard way to represent functional information, typically written as $f(x) = \text{expression in } x$ .
- $f$ represents the name of the function.
- $x$ represents the input variable.
- $f(x)$ represents the output of the function corresponding to the input $x$ .
Evaluating Linear Functions:
- For the function $f(x) = 3x + 2$ :
- To find $f(2)$ , replace every instance of $x$ with $(2)$ . Following the order of operations (PEMDAS), multiply first: $3 \times 2 = 6$ , then add: $6 + 2 = 8$ .
- To find $f(h)$ , replace $x$ with $h$ . The result is $3h + 2$ . Since $h$ is an unknown number, the expression cannot be simplified further.
Evaluating Power Functions:
- For the function $g(w) = w^3 + 1$ :
- Substituting $w = 1$ results in $1^3 + 1$ . Since $1 \times 1 \times 1 = 1$ , the result is $1 + 1 = 2$ .
- Substituting $w = -1$ results in $(-1)^3 + 1$ . Detailed calculation: $(-1) \times (-1) = 1$ , and $1 \times (-1) = -1$ . Therefore, $-1 + 1 = 0$ .
- Using parentheses when substituting negative numbers is critical to avoid calculation errors.
Variable Swapping and Imaginary Results:
- Functions can use different letters, such as $x(f) = \sqrt{f}$ . Here, the function is named $x$ and the input variable is $f$ .
- Evaluating $x(-1)$ yields $\sqrt{-1}$ . This is not a real number; it is an imaginary number denoted by $i$ .
- Evaluating $x(m + 1)$ results in $\sqrt{m + 1}$ .

When analyzing a function graph, movement is always tracked from left to right along the $x$ -axis.
Increasing: A function is increasing when the $y$ -values get larger (the graph rises) as one moves from left to right.
Decreasing: A function is decreasing when the $y$ -values get smaller (the graph falls) as one moves from left to right.
Constant: A function is constant when the $y$ -values stay exactly the same (the graph is a flat, horizontal line).
Warning Regarding Artificial Intelligence: AI tools often use Calculus to find maximums and minimums (taking derivatives and setting them to zero). In Algebra-based courses, students should use visual inspection of intervals rather than Calculus methods to avoid using advanced math that may not be required or allowed for specific assignments.

Absolute Maxima and Minima:
- The absolute maximum is the highest point on a graph. For example, if a function reaches its highest point at $x = 5$ with a $y$ -value of $1$ , the maximum is represented by the coordinate $(5, 1)$ .
- The absolute minimum is the lowest point. If a function is at its lowest level across an entire interval, such as from $x = -5$ to $x = -1$ , any $x$ in that interval (e.g., $x = -3, y = -6$ ) yields the minimum value.
Relative (Local) Maxima and Minima:
- Relative Minimum: A value $f(a)$ at $x = a$ is a relative minimum if $f(a)$ is the smallest function value relative to other points nearby. Visually, this is the bottom of a valley.
- Relative Maximum: A value $f(a)$ at $x = a$ is a relative maximum if $f(a)$ is the greatest function value relative to other points nearby. Visually, this is the top of a hill.
- For some graphs, there is no absolute maximum or minimum because the function continues toward positive or negative infinity (indicated by arrows). In these cases, only relative extrema exist.

Domain: The set of all possible $x$ -values (inputs) for which a function is defined.
- Certain functions have limited domains. For example, the square root function cannot accept negative numbers if the output must be a real number.
Range: The set of all resulting $y$ -values (outputs) that a function achieves.
Graphical Notation:
- Closed Circle: Indicates the point is defined and included in the set. Use brackets $[ \dots ]$ in interval notation.
- Open Circle: Indicates the function is not defined at that point, or the point is not included. Use parentheses $( \dots )$ in interval notation.
Example Analysis:
- If a graph starts with a closed circle at $x = -5$ and ends with an open circle at $x = 5$ , the Domain is $[-5, 5)$ .
- If the same graph spans $y$ -values from a closed circle at $y = -2$ to an open circle at $y = 5$ , the Range is $[-2, 5)$ .

Question: Does anyone see the single $x$ -value where this relation fails to be a function? (Asked regarding an upside-down point plot).
Response: The value is $2$ .
Question: What is the rule for PEMDAS in the expression $3(2) + 2$ ? Do we add the two first or multiply?
Response: Three times two occurs first.
Question: What is the result of one cubed ( $1^3$ )?
Response: One.
Question: What is negative one multiplied by itself? ( $-1 \times -1$ )
Response: Positive one.
Question: What is positive one times negative one?
Response: Negative one.