Thomas’ Calculus Fifteenth Edition Chapter 1 - Functions

Practice flashcards covering the definitions of functions, types of functions, graph transformations, and basic trigonometry from Thomas' Calculus Fifteenth Edition, Chapter 1.

Function

A rule from a set $D$ to a set $Y$ that assigns a single value in $Y$ to each $x$ in $D$, denoted as $y = f(x)$.

Independent Variable

The symbol $x$ representing the input value of a function $f$.

Dependent Variable

The symbol $y$, or output value, of a function $f$ at input $x$.

Vertical Line Test

A test stating that a curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the curve more than once.

Absolute Value Function

A piecewise-defined function with a domain of $(-\text{negative infinity}, \text{infinity})$ and a range of $[0, \text{infinity})$.

Greatest Integer Function

The function whose value at any number $x$ is the greatest integer less than or equal to $x$, often providing an integer floor for $x$, denoted as $y = \text{floor}(x)$.

Least Integer Function

The function whose value at any number $x$ is the smallest integer greater than or equal to $x$, often providing an integer ceiling for $x$, denoted as $y = \text{ceiling}(x)$.

Increasing Function

A function $f$ defined on an interval $I$ where $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ for points in $I$.

Decreasing Function

A function $f$ defined on an interval $I$ where $f(x_2) < f(x_1)$ whenever $x_1 < x_2$ for points in $I$.

Even Function

A function $f(x)$ where $f(-x) = f(x)$ for every $x$ in the domain, characterized by symmetry about the $y$-axis.

Odd Function

A function $f(x)$ where $f(-x) = -f(x)$ for every $x$ in the domain, characterized by symmetry about the origin.

Linear Function

A function of the form $f(x) = mx + b$, where $m$ and $b$ are constants.

Proportional

A relationship between variables $y$ and $x$ such that $y = kx$ for some nonzero constant $k$.

Polynomial

A function $p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $n$ is a nonnegative integer and coefficients are real constants.

Rational Function

A quotient or ratio $f(x) = \frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials and the domain is all real $x$ for which $q(x) \neq 0$.

Algebraic Function

Any function constructed from polynomials using algebraic operations such as addition, subtraction, multiplication, division, and taking roots.

Transcendental Function

Functions that are not algebraic, including trigonometric, inverse trigonometric, exponential, and logarithmic functions.

Composite Function

The function $(f \text{ composed with } g)$ defined by $(f \text{ circle } g)(x) = f(g(x))$.

Vertical Shift

A transformation that shifts the graph of $f$ up by $k$ units if $k > 0$ or down if $k < 0$, using the formula $y = f(x) + k$.

Horizontal Shift

A transformation that shifts the graph of $f$ left by $h$ units if $h > 0$ or right if $h < 0$, using the formula $y = f(x + h)$.

Radian Measure

The number $\theta = \frac{s}{r}$, representing the central angle where $r$ is the radius and $s$ is the arc length.

Periodic Function

A function $f(x)$ for which there is a positive number $p$ (the period) such that $f(x + p) = f(x)$ for every value of $x$.

ASTC Rule

A rule remembered by 'All Students Take Calculus' that identifies which trigonometric functions are positive in each of the four quadrants.

Law of Cosines

The geometric identity given by $c^2 = a^2 + b^2 - 2ab \times \text{cos}(\theta)$.

Sinusoid

The general sine function formula resulting from vertical and horizontal shifting and scaling.