Function Evaluation and Composition

Vocabulary flashcards covering function notation, evaluating inputs versus outputs, and function composition concepts based on the lecture.

Function Label ($f(x)$ or $g(x)$)

A naming notation swapped in place of $y$ used to distinguish and decipher between different equations or lines.

Input (Domain)

The numerical value or expression that replaces the independent variable $x$ inside a function.

Output (Range)

The resulting value produced by evaluating a function for a given input value.

Double Machine Model

A multi-step evaluation concept where an input is evaluated in an inner function first, and its output is then plugged into an outer function (e.g., evaluating $f(g(-3))$).

Evaluating $f(a)$

The process of directly replacing the variable $x$ with the number $a$ and performing arithmetic operations to find the output value.

Solving $f(x) = c$

The process of setting the entire function expression equal to the output value $c$ (replacing $f(x)$ or $y$) and isolating the variable $x$ to determine the input.

Function Composition ($f(g(x))$)

An operation where an entire function $g(x)$ is substituted into the variable $x$ of another function $f(x)$, followed by distributing and combining like terms.

Composition Notation ($g \circ h$)

A notation featuring a small open circle (a "mini o") representing $g(h(x))$, indicating that the function $h(x)$ is inserted into $g(x)$ rather than multiplied.

Inside-Out Evaluation

The procedural rule of starting with the innermost function or expression before working outward when evaluating composite functions.

FOIL Method (in Function Composition)

The algebraic technique used when expanding squared binomial inputs such as $(x + 3)^2$ into $x^2 + 6x + 9$ to ensure the middle term is not omitted.