General Mathematics Midterm Review: Functions, Rational Functions, Piecewise-Defined Functions, and Operations (Vocabulary)

Vocabulary flashcards covering key terms and definitions from functions, rational functions, piecewise-defined functions, and related operations as presented in the notes.

Functions

A relation that assigns each element of a set X to exactly one element of another set Y; written f: X → Y; input is the independent variable x and output is f(x) (often denoted y).

Relation

Any set of ordered pairs that connects elements from two sets; not all relations are functions.

Correspondence

Another term for pairing elements from two sets to form ordered pairs; synonymous with relation in this context.

Vertical Line Test

A graph represents a function if no vertical line intersects the graph more than once.

Domain

The set of all possible input values x for which a function is defined.

Range

The set of all possible output values y (or f(x)) produced by the function.

f(x) notation

The function rule written as f(x), where x is the input (independent variable) and f(x) is the output (dependent variable).

Independent Variable

The input variable, usually x, in a function.

Dependent Variable

The output variable, usually y or f(x), determined by the input.

Evaluating a Function

Substituting a value for x in f(x) and simplifying to obtain f(a).

Example of evaluating a function

If f(x) = 2x + 5, then f(2) = 9.

Polynomial Function

A function whose expression involves only non-negative powers of x (e.g., x^2, x^3).

Rational Function

A function that is the quotient of two polynomial functions P(x)/Q(x) with Q(x) ≠ 0.

Rational Expression

An algebraic fraction P(x)/Q(x) where P and Q are polynomials and Q ≠ 0.

Extraneous Roots

Solutions that arise during solving but are not allowed due to domain restrictions.

Least Common Denominator (LCD)

The least common denominator used to add or subtract rational expressions with different denominators.

Factor and Cancel

Factor numerator and denominator and cancel common factors when simplifying rational expressions.

Piecewise-Defined Function

A function defined by separate expressions on different sub-intervals of the domain.

Domain of a Piecewise Function

The union of the intervals for which each sub-function is defined; boundaries often shown with open/closed circles.

Graphing Piecewise Functions

Graph each sub-function on its respective interval and join with appropriate endpoints (solid/hollow circles as needed).

Absolute Value as Piecewise

Absolute value functions can be represented as piecewise-defined functions.

Applications of Piecewise Functions

Model real-world situations where a single formula does not describe all cases (e.g., salary with different rates up to a threshold).

Salary Function Example (Piecewise)

S(x) = 20,000 + 0.04x for 0 ≤ x ≤ 50,000; S(x) = 20,000 + 0.10x for x > 50,000.

Composition of Functions

(f ∘ g)(x) = f(g(x)); the inner function g is evaluated first; domain requires g(x) to be in dom(f).

Addition of Functions

(f + g)(x) = f(x) + g(x); domain is the intersection of dom(f) and dom(g).

Subtraction of Functions

(f − g)(x) = f(x) − g(x).

Multiplication of Functions

(f · g)(x) = f(x)g(x).

Division of Functions

(f / g)(x) = f(x) / g(x); requires g(x) ≠ 0; domain is the intersection where both functions are defined.

Polynomial Function

A function formed from polynomials; involves non-negative integer powers of x.

Rational Functions vs Rational Equations

A rational function is a quotient of polynomials (domain excludes zeros of the denominator). A rational equation is an equation involving rational expressions; not all such equations are functions.

Intercepts of a Function

x-intercepts are where f(x) = 0; y-intercept is where x = 0 (f(0)).

Asymptotes

Lines a graph approaches but does not cross; include vertical (x = a), horizontal (y = b), and slant (y = mx + b) asymptotes.

Vertical Asymptote

x-value where the function is undefined due to zero in the denominator (domain exclusion).

Horizontal Asymptote

y = L where the function tends to as x → ±∞; determined by degrees of numerator and denominator.

Slant (Oblique) Asymptote

A linear asymptote y = mx + b that occurs when the degree of the numerator is one more than the degree of the denominator.

Tables of Values (Rational Functions)

A method to graph rational functions by listing input-output pairs (x, f(x)).

Intercepts for Rational Functions

Finding x-intercepts by solving f(x) = 0 and y-intercept by evaluating f(0).

Domain Restrictions in Rational Functions

Denominator cannot be zero; exclude values that make the denominator zero.

Table of Values

A table listing x-values and corresponding f(x) values to aid graphing.

Real-World Modeling with Rational Functions

Using rational functions to model quantities like pricing, travel rates, or time dependent problems.

GEMDAS

Order of operations: Grouping, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right) used when evaluating functions.