College Algebra - Functions and Transformations Practice

Vocabulary terms related to the study of algebraic functions, their properties, compositions, and transformations as presented in the lecture notes.

Domain

The set of all possible input values for which a function is defined, often expressed using $(\text{interval notation})$.

Range

The set of all possible output values that a function results in, represented as an interval from the lowest to the highest $y$-value.

Function

A relation, such as $\{(-3,11), (-2,6), (0,2), (4,18)\}$, where each input value corresponds to exactly one output value.

Even Function

A function that satisfies the algebraic condition $f(-x) = f(x)$, such as $f(x) = 3x^2 + 4$.

Odd Function

A function that satisfies the algebraic condition $f(-x) = -f(x)$, such as $g(x) = x^3 - 4x$.

Increasing Interval

The span of $x$-values on a graph where the $y$-values increase as the $x$-values move from left to right.

Decreasing Interval

The span of $x$-values on a graph where the $y$-values decrease as the $x$-values move from left to right.

Absolute Maximum

The highest possible value that a function reaches over its entire domain.

Absolute Minimum

The lowest possible value that a function reaches over its entire domain.

Composition of Functions

The process of combining functions where the output of one function becomes the input of another, denoted as $(g \circ f)(x)$ or $g(f(x))$.

One-to-one Function

A function where every output value corresponds to exactly one unique input value, allowing for the existence of an inverse function.

Inverse Function

A function, denoted as $f^{-1}(x)$, that reverses the operation of the original function $f(x)$.

Horizontal Shift (Left)

A transformation that moves the graph of $f(x)$ to the left by $c$ units, represented algebraically as $f(x + c)$.

Vertical Shift (Upward)

A transformation that moves the graph of $f(x)$ up by $c$ units, represented algebraically as $f(x) + c$.

Average Rate of Change

The ratio of the change in the function values to the change in the input values over a specific interval, calculated as $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$.