1.9.1: Key Terms

Last updated
Save as PDF

Page ID: 116005

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Key Terms

absolute maximum: the greatest value of a function over an interval

absolute minimum: the lowest value of a function over an interval

absolute value equation: an equation of the form $| A | = B,$ with $B \geq 0;$ it will have solutions when $A = B$ or $A = - B$

absolute value inequality: a relationship in the form $| A | < B, | A | \leq B, | A | > B, or | A | \geq B$

average rate of change: the difference in the output values of a function found for two values of the input divided by the difference between the inputs

composite function: the new function formed by function composition, when the output of one function is used as the input of another

decreasing function: a function is decreasing in some open interval if $f (b) < f (a)$ for any two input values $a$ and $b$ in the given interval where $b > a$

dependent variable: an output variable

domain: the set of all possible input values for a relation

even function: a function whose graph is unchanged by horizontal reflection, $f (x) = f (- x),$ and is symmetric about the $y -$ axis

function: a relation in which each input value yields a unique output value

horizontal compression: a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant $b > 1$

horizontal line test: a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once

horizontal reflection: a transformation that reflects a function’s graph across the y-axis by multiplying the input by $−1$

horizontal shift: a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input

horizontal stretch: a transformation that stretches a function’s graph horizontally by multiplying the input by a constant $0 < b < 1$

increasing function: a function is increasing in some open interval if $f (b) > f (a)$ for any two input values $a$ and $b$ in the given interval where $b > a$

independent variable: an input variable

input: each object or value in a domain that relates to another object or value by a relationship known as a function

interval notation: a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion

inverse function: for any one-to-one function $f (x),$ the inverse is a function $f^{- 1} (x)$ such that $f^{- 1} (f (x)) = x$ for all $x$ in the domain of $f;$ this also implies that $f (f^{- 1} (x)) = x$ for all $x$ in the domain of $f^{- 1}$

local extrema: collectively, all of a function's local maxima and minima

local maximum: a value of the input where a function changes from increasing to decreasing as the input value increases.

local minimum: a value of the input where a function changes from decreasing to increasing as the input value increases.

odd function: a function whose graph is unchanged by combined horizontal and vertical reflection, $f (x) = - f (- x),$ and is symmetric about the origin

one-to-one function: a function for which each value of the output is associated with a unique input value

output: each object or value in the range that is produced when an input value is entered into a function

piecewise function: a function in which more than one formula is used to define the output

range: the set of output values that result from the input values in a relation

rate of change: the change of an output quantity relative to the change of the input quantity

relation: a set of ordered pairs

set-builder notation: a method of describing a set by a rule that all of its members obey; it takes the form ${x | statement about x}$

vertical compression: a function transformation that compresses the function’s graph vertically by multiplying the output by a constant $0 < a < 1$

vertical line test: a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once

vertical reflection: a transformation that reflects a function’s graph across the x-axis by multiplying the output by $−1$

vertical shift: a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output

vertical stretch: a transformation that stretches a function’s graph vertically by multiplying the output by a constant $a > 1$

Search

Text Color

Text Size

Margin Size

Font Type

Key Terms