### Key Terms

- absolute value function
- $f\left(x\right)=\{\begin{array}{c}\text{\u2212}x,x<0\\ x,x\ge 0\end{array}$

- 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 $f(\text{\u2212}x)=f(x)$ 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 ${e}^{\text{\u2212}x}$

- 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 $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ for some base $b>0,b\ne 1$ such that $y={\mathrm{log}}_{b}(x)$ 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 $\text{ln}\phantom{\rule{0.1em}{0ex}}x={\text{log}}_{e}x$

- 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 $f(\text{\u2212}x)=\text{\u2212}f(x)$ 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 $f\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\text{\u2026}+{a}_{1}x+{a}_{0}$

- 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 $(\text{\u2212}x,\text{\u2212}y)$ 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 $(\text{\u2212}x,y)$ 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 $f,f(x)=0$