Review of Functions: practice problems

Key terms · 17

absolute value function

$f (x) = \{\begin{array}{l} \text{-} x, x < 0 \\ x, x \ge 0 \end{array}$

composite function

given two functions $f$ and $g,$ a new function, denoted $g \circ f,$ such that $(g \circ f) (x) = g (f (x))$

decreasing on the interval $I$

a function decreasing on the interval $I$ if, for all $x_1, x_2 \in I, f (x_1) \ge f (x_2)$ if $x_1 < x_2$

dependent variable

the output variable for a function

domain

the set of inputs for a function

even function

a function is even if $f (\text{-} x) = f (x)$ for all $x$ in the domain of $f$

function

a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output

graph of a function

the set of points $(x, y)$ such that $x$ is in the domain of $f$ and $y = f (x)$

increasing on the interval $I$

a function increasing on the interval $I$ if for all $x_1, x_2 \in I, f (x_1) \le f (x_2)$ if $x_1 < x_2$

independent variable

the input variable for a function

odd function

a function is odd if $f (\text{-} x) = \text{-} f (x)$ for all $x$ in the domain of $f$

range

the set of outputs for a function

symmetry about the origin

the graph of a function $f$ is symmetric about the origin if $(\text{-} x, \text{-} y)$ is on the graph of $f$ whenever $(x, y)$ is on the graph

symmetry about the y-axis

the graph of a function $f$ is symmetric about the $y$-axis if $(\text{-} x, y)$ is on the graph of $f$ whenever $(x, y)$ is on the graph

table of values

a table containing a list of inputs and their corresponding outputs

vertical line test

given the graph of a function, every vertical line intersects the graph, at most, once

zeros of a function

when a real number $x$ is a zero of a function $f, f (x) = 0$