characteristics of functions (3.2) and transformations (3.3)

continuous function

A function whose graph is an unbroken line or curve with no gaps or breaks.

continuous everywhere

domain is all reals, can be drawn without picking up your pencil

continuous on domain

restricted domains, but can still be drawn w/out picking up your pencil/any breaks

common functions that are continuous everywhere

- constant
- identity
- absolute value
- quadratics
- cubic
- cube root
- exponential decay
- exponential growth

common functions that are continuous over their entire domain

- square root
- reciprocal
- logarithmic functions (both 0 < b < 1 and b \> 1)
- signum

Continuity at a point (x \= c)

function value must exist; when sketching the function, your pencil passes through that point w/out having to pick it up

continuity over an interval

function must be continuous at every point in the interval. if the interval is closed, you must be able to sketch it from beginning to end w/out having to pick up your pencil.

discontinuous function

A function whose graph has one or more jumps, breaks, or holes; NOT continuous at all x-values

types of discontinuities

regional ("gap"), infinite, point ("hole" or "removable discontinuity"), jump

regional discontinuity/gap

usually caused by even roots of a negative number or piecewise
-\> has x-values which do not have defined f(x) values

infinite discontinuity

A value of x that creates a vertical asymptote on a function.

Point Discontinuity ("Hole")

the point at which a mathematical function is no longer continuous; hole/open circle on the graph where the left and right sides of the function meet but the point itself does not exist

jump discontinuity

A graph that has discontinuity where the function moves to a different y-value and then continues. It cannot be filled in with just a point

x-intercept

the x-coordinate of a point where a graph crosses the x-axis
- known as solutions, zeroes, or roots

y-intercept

the y-coordinate of a point where a graph crosses the y-axis

positive function values

Piece(s) of the graph which falls above the x-axis and whose
y-coordinates are positive real numbers.
- solution to the inequality f(x) \> 0 gives the set of x-values for which the function has positive y-values.

negative function values

Piece(s) of the graph which falls below the x-axis and whose
y-coordinates are negative real numbers.
- Solution to the inequality f(x) < 0 gives the set of x-values for which the function has negative y-values.

critical numbers for sign change

x-values where the graph may switch from positive y values to negative y-values or vice versa.
- The critical numbers are the zeroes, restrictions and any discontinuities that are not restrictions.

difference quotient

f(-x) = f(x)

even function, y-axis symmetry

f(-x) = -f(x)

odd function, origin symmetry