Domain & Range Domain (D): is all the x values Range (R): is all the y values Must write D and R in interval notation To find domain algebraically set the denominator or radical = 0, then solve you must check to see if your answer works by checking the 3 following points: #, if it is a true statement it is part of your domain, if it’s false it isn’t in your domain

Domain and range cont’d Graphically For domain: look at the graph and determine where is the lowest x value, where is the highest x value and are there any missing For range: look at the graph and determine where is the lowest y value, where is the highest y value and are there any missing

Continuity/discontinuity Continuous: a function is said to have continuity if there are no breaks in the graph for example: Discontinuous: a function is said to have discontinuity if there are any breaks in the graph, there are 3 types

Discontinuity Removable: there is a hole in the graph, this happens when you factor and cancel Infinite: there is an entire line missing from the graph

Discontinuity cont’d Jump: this happens when the graph jumps over points

Increasing/decreasing/constant To write increasing/decreasing/constant intervals only look at the x values Don’t forget to use open or closed brackets as needed

Boundedness Unbounded: is a function that has no turning points, for ex: a straight line Bounded: this is when there is at least 1 turning point 3 types of bounded: Bounded above: it has a maximum point

Bounded cont’d Bounded below: the function has a minimum point Bounded: the function has at least 1 minimum & maximum

Local & Absolute Extrema Extrema is another name for maximum or minimum Local maximum: is a value of f(c) that is greater than or equal to all range values of f on some open interval Local minimum: is a value of f(c) that is less than or equal to all range values of f on some open interval Absolute maximum: is the highest point Absolute minimum: is the lowest point If a function is bounded, you must identify the absolute maximum and/or minimum

Symmetry Even functions: are symmetric over the y-axis Odd functions: are symmetric over the origin (y-axis and x-axis) Neither: are not symmetric at all Examples evenoddneither

Asymptotes Vertical Asymptote (VA): is a vertical line that the graph does not touch or cross To find: 1) factor and cancel if you can (if you cancel anything there is a hole at that point) 2) set the denominator = 0 and solve Must write in the form x = # If the function is continuous there is no VA

Asymptotes Horizontal Asymptotes (HA): a horizontal line that the graph does not cross or touch To find: 3 conditions if f(x) = ax n bx m 1) if n > m, there is no HA 2) if n = m, then the HA is y = a/b 3) if n < m, the HA is y = 0 (always look at the highest exponents, to find the HA, the function must be in unfactored form)

End Behavior End behavior describes where f(x) (the y values) is going as the x values get bigger or smaller Written using limits To write: lim f(x) = & lim f(x) x -∞ x ∞ (read the limit of f(x) as x approaches -∞ and the limit of f(x) as x approaches ∞)