1.9.1: Key Terms

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May 28, 2023
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- 1.9: Chapter Review
- 1.9.2: Key Equations

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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 $a > 1$

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