3-5 Continuity and End Behavior

Discontinuous: A function is discontinuous if you can not trace it without lifting your pencil.

Infinite Discontinuity – as the graph of f(x) approaches a given value of x, lf(x)l becomes increasingly greater Jump Discontinuity – The graph of f(x) stops and then begins again with an open circle at a different range value for a given value of the domain. Point Discontinuity – When there is a value in the domain for which f(x) is undefined, but the pieces of the graph match up Everywhere Discontinuous – A function that is impossible to graph in the real number system is said to be everywhere discontinuous

Continuous Test A function is continuous at x = c if it satisfies the following conditions: 1) The function is defined at c, in other words F(c) exists 2) The function approaches the same y-value on the left and right sides of x = c;

Determine if the function is continuous at the given point f(x) = 3x2 +7; x = 1 2) F(x)= 𝑥−2 𝑥 2 −4 ; x = 2 3) F(x)= 1 𝑥 𝑖𝑓 𝑥>1 𝑥 𝑖𝑓 𝑥≤1 ; x=1

Continuity on an Interval A function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval. 4) 𝑓 𝑥 = 3𝑥−2 𝑖𝑓 𝑥>2 2−𝑥 𝑖𝑓 𝑥≤2 from -2<x<3

End Behavior The behavior of f(x) as lxl becomes larger and larger (where is the graphing going?) Typically, a graph approaches ∞ 𝑜𝑟 −∞ We write the answer like "𝑓(𝑥)→∞ 𝑎𝑠 𝑥→∞”

Describe the end behavior of: 5) F(x) = -2x3 6) F(x) = -x3 +x2 –x+5

Increasing, Decreasing, and Constant Functions A function f is increasing on an interval I if and only if for every a and b contained in I, f(a) < f(b) whenever a < b. A function f is decreasing on an interval I if and only if for every a and b contained in I, f(a)> F(b) whenever a < b. A function f remains constant on an interval I if and only if for every a and b contained in I, f(a) = f(b) whenever a < b

Determine the intervals each function is increasing or decreasing. 7)Graph f(x) = 3- (x-5)2 8) Graph f(x) = ½ lx+3l – 5 9)Graph f(x) = 2x3 + 3x2- 12x+ 3