Sec 2.1-2.2 Functions and Domain/Range Vocabulary

Vocabulary flashcards covering key concepts from Sec 2.1–2.2 on functions, domains/ranges, graphs, and basic function types.

Domain

The set of all inputs (x-values) for a relation or function; may be stated explicitly, e.g., the domain of f(x) = x^2 with 0 ≤ x ≤ 5 is [0,5].

Range

The set of all outputs (y-values) of a relation or function; for the example above, the range is [0,25].

Graph

The representation of a function in the xy-plane; a graph represents a function if it passes the Vertical Line Test.

Vertical Line Test (VLT)

A graph is a function if every vertical line intersects the graph at most once.

Function

A relation where each input has exactly one output; the domain is the input set and the range is the output set.

Independent Variable

The input variable (usually x) in a function; the domain consists of possible inputs.

Dependent Variable

The output variable (usually y) in a function; the range consists of possible outputs.

Function Notation

f(x) denotes the output of function f for input x; it is not f times x.

Evaluating a Function

Substitute a value for x in f(x) to obtain f(a) = value.

Placeholder

In f(x), x acts as a placeholder to be replaced by a value when evaluating the function.

Piecewise Function

A function defined differently for different parts of the domain.

Domain in Interval Notation

Expressing the domain as intervals, such as [a,b], (a,b], etc.

Range in Interval Notation

Expressing the range as intervals.

Linear Function

A function of the form f(x) = mx + b.

Polynomial Function

A function of the form f(x) = x^n, where n is a nonnegative integer.

Root Function

Functions involving roots; examples include f(x) = sqrt(x) and f(x) = x^3.

Rational Function

A function of the form f(x) = p(x)/q(x) where p and q are polynomials and q(x) ≠ 0.

Exponential Function

A function of the form f(x) = b^x where b > 0 and b ≠ 1.

Absolute Value Function

f(x) = |x|, a V-shaped graph; nonnegative outputs.

Complex Number

A number of the form a + bi, where i is the imaginary unit with i^2 = -1; a is the real part and bi is the imaginary part.

Imaginary Unit i

i = sqrt(-1); i^2 = -1.

Conjugate

For a + bi, the conjugate is a - bi.

Real Part and Imaginary Part

In a complex number a + bi, a is the real part and bi is the imaginary part.

Simplifying Complex Numbers

Use i^2 = -1 and combine like terms as with real numbers.

Inequality

A relation using

Sign Change in Inequalities

When you multiply or divide by a negative number, you must flip the inequality sign.

Absolute Value Inequality

Isolate the absolute value and split into two inequalities; combine with AND if the original uses < or ≤, OR if it uses > or ≥.

Example of No Solution for |x| < a

If a ≤ 0, the inequality |x| < a has no solution.

Domain Restrictions for Radicals (Even Index)

The radicand must be ≥ 0 for real numbers (or >0 if the radical is in the denominator).

Domain Restrictions for Denominators

The domain excludes x-values that make a denominator zero.

Real Life Limitations on Domain

Some quantities cannot take certain values (e.g., time cannot be negative); domains may be restricted by context.