Calculus 1: Relations and Functions

Vocabulary flashcards covering core definitions, mathematical properties, and graphs of relations and functions from Calculus 1.

Relation

A set of ordered pairs $(x, y)$ in which the order matters and where $x$ corresponds to $y$ or $y$ depends on $x$.

Gottfried Leibniz

The German mathematician (1646–1716) who first used the term "function" in mathematics and co-invented calculus.

Function

A rule that assigns to each element $x$ in a set $A$ exactly one element called $f(x)$ in a set $B$.

Domain

The set of all admissible values of $x$ (inputs) for a given function.

Range

The set of all resulting values of $y$ (outputs) for a given function.

One-to-One Function

A relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component, and each value of the second component corresponds to exactly one value of the first component.

Functional Notation

The notation $y = f(x)$, read as "f of x", used to emphasize that $y$ is a function of $x$ or $y$ depends on $x$.

Leonhard Euler

The mathematician (1707–1783) who first introduced the $f(x)$ notation for functions.

Algebraic Function

A function formed by a finite number of algebraic operations on identity and constant functions.

Polynomial Function

An algebraic function consisting of terms with non-negative integer powers of $x$; examples include constant, linear, quadratic, and cubic functions.

Constant Function

A polynomial function of degree 0 defined by $y = c$, having a domain of all real numbers $\{x \mid x \in \mathbb{R}\}$ and range $\{y \mid y = c\}$.

Linear Function

A polynomial function of degree 1 defined by $y = mx + b$ ($m \neq 0$), having a domain of all real numbers and range of all real numbers.

Identity Function

A particular linear function defined by $y = x$.

Quadratic Function

A polynomial function of degree 2 defined by $y = ax^2 + bx + c$ where $a \neq 0$.

Cubic Function

A polynomial function of degree 3 defined by $y = ax^3 + bx^2 + cx + d$ where $a \neq 0$.

Rational Function

A function expressed as the quotient of two polynomial functions $y = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$.

Power Function

A function defined by $y = x^n$, where $n$ is a constant; includes radical functions ($y = x^{1/n}$) and reciprocal functions ($y = x^{-n}$).

Piece-wise Function

A function defined using more than one expression across different intervals of its domain, such as the absolute value function.

Transcendental Function

A function that is not algebraic, including trigonometric, logarithmic, and exponential functions.

Trigonometric Functions

Functions defined as the ratios of lengths of sides of a right triangle, consisting of sine, cosine, tangent, cotangent, secant, and cosecant.

Logarithmic Function

A function defined by $y = \log_b(x)$, where $b > 0$ and $b \neq 1$; it is the inverse function of an exponential function.

Exponential Function

A function defined by $y = b^x$, where $b > 0$ and $b \neq 1$; it is the inverse function of a logarithmic function.

Graph of Exponential Function $f(x) = b^x$

An exponential curve passing through $(0, 1)$ with domain of all real numbers and range of all positive numbers ($y > 0$).

Graph of Natural Exponential Function $f(x) = e^x$

An exponential curve with base $e \approx 2.718$, passing through $(0, 1)$ and $(1, e)$.

Graph of Logarithmic Functions $f(x) = \ln(x)$ and $f(x) = \log(x)$

Curves passing through $(1, 0)$ with domain of all positive real numbers ($x > 0$) and range of all real numbers.

Graph of Sine Function $f(x) = \sin(x)$

A periodic wave passing through $(0, 0)$, with domain of all real numbers and range $-1 \le y \le 1$.

Graph of Cosine Function $f(x) = \cos(x)$

A periodic wave passing through $(0, 1)$, with domain of all real numbers and range $-1 \le y \le 1$.

Graph of Tangent Function $f(x) = \tan(x)$

Periodic curves with vertical asymptotes at $x = \frac{\pi}{2} + k\pi$, domain of all real numbers except those asymptotes, and range of all real numbers.

Graph of Cotangent Function $f(x) = \cot(x)$

Periodic curves with vertical asymptotes at $x = k\pi$, domain of all real numbers except $k\pi$, and range of all real numbers.

Graph of Secant Function $f(x) = \sec(x)$

Periodic U-shaped curves with vertical asymptotes at $x = \frac{\pi}{2} + k\pi$ and range $(-\infty, -1] \cup [1, +\infty)$.

Graph of Cosecant Function $f(x) = \csc(x)$

Periodic U-shaped curves with vertical asymptotes at $x = k\pi$ and range $(-\infty, -1] \cup [1, +\infty)$.

Vertical Line Test

A visual method used to determine if a relation's graph is a function; if any vertical line intersects the graph at more than one point, the graph does not represent a function.

Algebra of Functions

Operations combining function values: addition $(f+g)(x) = f(x)+g(x)$, subtraction $(f-g)(x) = f(x)-g(x)$, multiplication $(f \cdot g)(x) = f(x)g(x)$, and division $(f/g)(x) = \frac{f(x)}{g(x)}$ ($g(x) \neq 0$).

Composite Function

A function formed by using the output of one function as the input for a second function, denoted by $(f \circ g)(x) = f(g(x))$.

Even Function

A function $f$ that satisfies $f(-x) = f(x)$ for every $x$ in its domain.

Odd Function

A function $f$ that satisfies $f(-x) = -f(x)$ for every $x$ in its domain.

Inverse Function

A function formed by reversing the coordinates of each ordered pair of a one-to-one function, denoted by $f^{-1}(x)$, where $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Horizontal Line Test

A test used to determine if a function's graph represents a one-to-one function; if any horizontal line intersects the graph at more than one point, the function is not one-to-one.