Standard Achievement Admission Test (SAAT) - Comprehensive Study Notes

Function Classification

A function is characterized by the relationship between its domain (input values) and range (output values). Understanding how a function behaves is crucial in various fields of mathematics and applied sciences.
    Increasing Functions

A function is considered increasing on an interval if, for any two numbers $x_1$ and $x_2$ within that interval, the condition x_1 < x_2 implies f(x_1) < f(x_2). On a graph, the curve moves upwards from left to right.
    Decreasing Functions

Conversely, a function is decreasing on an interval if, for any two numbers $x_1$ and $x_2$ in that interval, the condition x_1 < x_2 implies f(x_1) > f(x_2). On a graph, the curve moves downwards from left to right.
    Constant Functions

A function is constant on an interval if for any two numbers $x_1$ and $x_2$ in the interval, the output remains unchanged, or $f(x_1) = f(x_2)$. The graph will appear as a horizontal line.

Critical Points and Extreme Values

Critical points are where the function alters its behavior, typically shifting from increasing to decreasing, or vice versa. These points can form either:
   

• Tops (maximums): The highest points in the function's graph.

• Bottoms (minimums): The lowest points in the function's graph.     Types of Extremes

• Absolute Maximum: The highest value across the entire domain of the function.

• Local Maximum: A point that is higher than its immediate neighbors, but not necessarily the highest overall.

• Absolute Minimum: The lowest value across the entire domain of the function.

• Local Minimum: A point that is lower than its immediate neighbors, but not necessarily the lowest overall.     Interval Notation for Behavior

To describe the behavior of a function over intervals, we use interval notation, which employs open brackets $(,)$. It's essential to note that a function cannot be labeled as increasing or decreasing at just a single point.     Example Behaviors

For instance, consider the function $f(x)$:
   

• It increases in $(- ext{infinity}, -5)$

• Remains constant in $(-5, 0)$

• Decreases in $(0, ext{infinity})$     The following flowchart summarizes these characteristics: