Key Terms

absolute value function

$f\left(x\right)=\{\begin{array}{c}\text{−}x,x<0\\ x,x\ge 0\end{array}$

algebraic function

a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable $x$

base

the number $b$ in the exponential function $f\left(x\right)=b^x$ and the logarithmic function $f\left(x\right)={\text{log}}_{b}x$

composite function

given two functions $f$ and $g,$ a new function, denoted $g\circ f,$ such that $\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$

cubic function

a polynomial of degree 3; that is, a function of the form $f\left(x\right)=a{x}^3+b{x}^2+cx+d,$ where $a\ne 0$

decreasing on the interval $I$

a function decreasing on the interval $I$ if, for all $x_1,x_2\in I,f(x_1)\ge f(x_2)$ if $x_1<x_2$

degree

for a polynomial function, the value of the largest exponent of any term

dependent variable

the output variable for a function

domain

the set of inputs for a function

even function

a function is even if $f(\text{−}x)=f(x)$ for all $x$ in the domain of $f$

exponent

the value $x$ in the expression $b^x$

function

a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output

graph of a function

the set of points $(x,y)$ such that $x$ is in the domain of $f$ and $y=f(x)$

horizontal line test

a function $f$ is one-to-one if and only if every horizontal line intersects the graph of $f,$ at most, once

hyperbolic functions

the functions denoted $\text{sinh},\text{cosh},\text{tanh},\text{csch},\text{sech},$ and $\text{coth},$ which involve certain combinations of $e^x$ and $e^{\text{−}x}$

increasing on the interval $I$

a function increasing on the interval $I$ if for all $x_1,x_2\in I,f(x_1)\le f(x_2)$ if $x_1<x_2$

independent variable

the input variable for a function

inverse function

for a function $f,$ the inverse function $f^{\mathrm{-1}}$ satisfies $f^{\mathrm{-1}}\left(y\right)=x$ if $f\left(x\right)=y$

inverse hyperbolic functions

the inverses of the hyperbolic functions where $\text{cosh}$ and $\text{sech}$ are restricted to the domain $\left[0,\infty \right);$ each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function

inverse trigonometric functions

the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions

linear function

a function that can be written in the form $f\left(x\right)=mx+b$

logarithmic function

a function of the form $f\left(x\right)={\log }_b\left(x\right)$ for some base $b>0,b\ne 1$ such that $y={\log }_b(x)$ if and only if $b^y=x$

mathematical model

A method of simulating real-life situations with mathematical equations

natural exponential function

the function $f\left(x\right)=e^x$

natural logarithm

the function $\text{ln}x={\text{log}}_{e}x$

number e

as $m$ gets larger, the quantity $(1+{(1\text{/}m))}^m$ gets closer to some real number; we define that real number to be $e;$ the value of $e$ is approximately $2.718282$

odd function

a function is odd if $f(\text{−}x)=\text{−}f(x)$ for all $x$ in the domain of $f$

one-to-one function

a function $f$ is one-to-one if $f\left(x_1\right)\ne f\left(x_2\right)$ if $x_1\ne x_2$

periodic function

a function is periodic if it has a repeating pattern as the values of $x$ move from left to right

piecewise-defined function

a function that is defined differently on different parts of its domain

point-slope equation

equation of a linear function indicating its slope and a point on the graph of the function

polynomial function

a function of the form $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\text{…}+a_{1}x+a_0$

power function

a function of the form $f\left(x\right)=x^n$ for any positive integer $n\ge 1$

quadratic function

a polynomial of degree 2; that is, a function of the form $f\left(x\right)=a{x}^2+bx+c$ where $a\ne 0$

radians

for a circular arc of length $s$ on a circle of radius 1, the radian measure of the associated angle $\theta$ is $s$

range

the set of outputs for a function

rational function

a function of the form $f\left(x\right)=p(x)\text{/}q\left(x\right),$ where $p\left(x\right)$ and $q\left(x\right)$ are polynomials

restricted domain

a subset of the domain of a function $f$

root function

a function of the form $f\left(x\right)=x^{1\text{/}n}$ for any integer $n\ge 2$

slope

the change in y for each unit change in x

slope-intercept form

equation of a linear function indicating its slope and y-intercept

symmetry about the origin

the graph of a function $f$ is symmetric about the origin if $(\text{−}x,\text{−}y)$ is on the graph of $f$ whenever $(x,y)$ is on the graph

symmetry about the y-axis

the graph of a function $f$ is symmetric about the $y$-axis if $(\text{−}x,y)$ is on the graph of $f$ whenever $(x,y)$ is on the graph

table of values

a table containing a list of inputs and their corresponding outputs

transcendental function

a function that cannot be expressed by a combination of basic arithmetic operations

transformation of a function

a shift, scaling, or reflection of a function

trigonometric functions

functions of an angle defined as ratios of the lengths of the sides of a right triangle

trigonometric identity

an equation involving trigonometric functions that is true for all angles $\theta$ for which the functions in the equation are defined

vertical line test

given the graph of a function, every vertical line intersects the graph, at most, once

zeros of a function

when a real number $x$ is a zero of a function $f,f(x)=0$