Section 5 – Continuity and End Behavior Chapter 3 Section 5 – Continuity and End Behavior

Discontinuity Most of the graphs that we’ve seen so far have been smooth, continuous functions. When a function is discontinuous, it is because it looks “broken” or untraceable without lifting your pencil.

Types of Discontinuity Infinite Dscnty Jump Dscnty Point Dscnty

Determining Continuity at a Given Point The Continuity Test: To see if a function is continuous at a given point: (x=a) Does f(a) exist??? Plug the given value of x into the function and see if it is solvable. [cannot have 4 0 zero denominator, cannot have −1 imaginary numbers] X=1 Exsists! X=2 is undefined

Determining Continuity at a Given Point The Continuity Test: Does f(x) approach the same value from both sides? [as x increases toward a or decreases toward a, does it approach the same thing?] The actual y-value that f(x) approaches from both side IS f(x) (rather than approaching a point-discontinuity) Discontinuities will usually mean x in the denominator of a fraction and/or x inside an even numbered root [like: 𝟏 𝒙 or 𝟒 𝒙−𝟐 ]

Determining Continuity at a Given Point EX 1: Determine whether f(x)=3x2+7 is continuous at x=1. EX 2: Determine whether g(x)=(x-2)/(x2-4) is continuous at x=2 EX 3: Determine whether f(x)= 1/x,x>1 x, x≤1 is continuous at x=1

End Behavior Another tool for analyzing functions is to see the end behavior of a function, which is to say as x approaches infinity and negative infinity.

End Behavior for Polynomial Functions

Assignment Chapter 3, Section 5 pgs 165-166 #6,8,12-28E,33