Functions and Their Graphs

Question-and-Answer flashcards covering relations, functions, and various types of function graphs.

What is a relation in mathematics?

A relation is a connection between elements of one or more sets.

How is a function different from a general relation?

A function is a relation in which no element of the domain (x-value) is repeated; every input has exactly one output.

When examining ordered pairs, how can you tell if the relation is a function?

If no x-value (domain element) appears more than once, the relation is a function.

List at least three common representations of relations or functions.

Ordered pairs, tables of values, mapping diagrams, graphs, phrases, or sentences.

What graphical feature identifies a linear function?

Its graph is an always-slant straight line.

Does a typical linear function pass through the origin?

No; a linear function such as f(x)=5x−3 generally does not pass through the origin.

What distinguishes an identity function from a general linear function?

An identity function is a slant line that passes through the origin, e.g., f(x)=5x.

What type of line represents a constant function on a graph?

A constant function is represented by a horizontal line.

What is the degree and basic graph shape of a quadratic function?

Degree 2; its graph is a parabola that opens upward or downward.

Give the general algebraic form of a quadratic function.

f(x) = ax² + bx + c

What is the degree of a cubic function, and what is its general form?

Degree 3; f(x) = ax³ + bx² + cx + d

How can you recognize an absolute value function algebraically?

It contains an absolute-value symbol, e.g., f(x)=a|x−h|+k.

What key feature identifies a square root function?

It involves the square root of a polynomial, e.g., f(x)=√x.

Define a rational function in simplest terms.

A function expressed as a ratio of two polynomials, f(x)=p(x)/q(x), which may produce decimal or fractional outputs.