### 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 $\left\{x\right|\phantom{\rule{0.8em}{0ex}}\text{statementabout}x\}$

- 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 $a>1$