Functions and Graphs Vocabulary

Vocabulary flashcards covering key concepts from the lecture notes on functions, domains, ranges, intercepts, symmetry, and common types of functions.

Vertical line test

A method to determine if a graph represents a function: any vertical line should intersect the graph at most once for every x in the domain.

Function

A relation in which every input value has exactly one output value.

Domain

The set of all input values (x-values) for which a function is defined.

Range

The set of all output values (y-values) produced by a function.

X-intercept

The point(s) where the graph crosses the x-axis (y = 0), written as (a, 0).

Y-intercept

The point(s) where the graph crosses the y-axis (x = 0), written as (0, b).

Even function

A function symmetric about the y-axis; f(-x) = f(x).

Odd function

A function symmetric about the origin; f(-x) = -f(x).

Identity function

f(x) = x; the output equals the input.

Constant function

f(x) = c; outputs do not depend on x; graph is a horizontal line.

Absolute value function

f(x) = |x|; domain all real numbers; range [0, ∞); V-shaped, even; increases for x>0 and decreases for x<0; minimum at x=0.

Square root function

f(x) = sqrt(x); domain x ≥ 0; range y ≥ 0; increasing; intercept at (0,0).

Cube root function

f(x) = x^(1/3); domain all real numbers; range all real numbers; increasing; intercept at (0,0).

Piecewise function

A function defined by different expressions depending on the input; uses conditional pieces or if statements.

Denominator restriction

Values of x that make a function undefined (e.g., when the denominator is zero), so they are not in the domain.

Increasing

A property where as x increases, the output y increases; the graph rises left to right.

Decreasing

A property where as x increases, the output y decreases; the graph falls left to right.

Absolute maximum

The largest y-value attained by the function on its entire domain.

Absolute minimum

The smallest y-value attained by the function on its entire domain.

Complex numbers

Numbers of the form a + bi; i is the imaginary unit with i^2 = -1 (e.g., sqrt(-1) = i).

Interval notation for domain

A way to express the domain, e.g., (-∞, ∞) for all real numbers, or [0, ∞) for nonnegative inputs.

Interval notation for range

A way to express the range, e.g., [0, ∞) for nonnegative outputs.

X-intercept notation

The x-intercept is the point where the graph crosses the x-axis, with y = 0; written as (a, 0).

Y-intercept notation

The y-intercept is the point where the graph crosses the y-axis, with x = 0; written as (0, b).

f(x) notation

The output value of a function corresponding to the input x; often used to describe the mapping y = f(x).