Key Terms

Key Terms

absolute maximum

the greatest value of a function over an interval

absolute minimum

the lowest value of a function over an interval

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 $f\left(b\right)<f\left(a\right)$ 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 $y\text{-}$ 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 $\mathrm{-1}$

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 $f\left(b\right)>f\left(a\right)$ 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 $f^{-1}\left(f\left(x\right)\right)=x$ for all $x$ in the domain of $f;$ this also implies that $f\left(f^{-1}\left(x\right)\right)=x$ 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 $\left\{x\right|\text{statement about }x\}$

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 $\mathrm{-1}$

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 $a>1$