Algebra and Trigonometry

# 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$f( b )for any two input values$a a$and$b b$in the given interval where$b>a 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,$f(x)=f(−x), f(x)=f(−x),$and is symmetric about the $y- y-$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$b>1 b>1$
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$−1 −1$
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$0
increasing function
a function is increasing in some open interval if$f( b )>f( a ) f( b )>f( a )$for any two input values$a a$and$b b$in the given interval where$b>a 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$f(x), f(x),$the inverse is a function$f −1 (x) f −1 (x)$such that$f −1 ( f( x ) )=x f −1 ( f( x ) )=x$for all$x x$in the domain of$f; f;$this also implies that$f( f −1 ( x ) )=x f( f −1 ( x ) )=x$for all$x x$in the domain of$f −1 f −1$
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,$f(x)=−f(−x), f(x)=−f(−x),$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
vertical compression
a function transformation that compresses the function’s graph vertically by multiplying the output by a constant$0
vertical line test
a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once
vertical reflection
a transformation that reflects a function’s graph across the x-axis by multiplying the output by$−1 −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$a>1 a>1$