Continuity & Discontinuity Increasing & Decreasing Of Functions

Objective SWBAT: – Identify whether a function is continuous or discontinuous – Identify the types of discontinuity – Identify when a function is increasing, decreasing, or constant with the intervals respectively.

Definition of Continuity A function is continuous on an open interval (a, b) if it is continuous on each point in the interval. A function that is continuous on the entire real line is everywhere continuous. f(x) is continuous on (-3,2)

A function is continuous if you can draw it in one motion without picking up your pencil.

jump infinite Nonremovable Discontinuities: Removable Discontinuities: (You can fill the hole.)

“Discussing Continuity” Continuous or discontinuous? If discontinuous – Removable or nonremovable discontinuity? – At what x-value is the discontinuity? 6

Continuity by Function Type Polynomials are everywhere continuous Sine and Cosine are everywhere continuous Rational functions and other trig functions are continuous except at x- values where their denominators equal zero. – “Removable” discontinuity if factoring and canceling “removes” the zero in the denominator – “Non-removable” otherwise. (Recall that vertical asymptotes occur where numerator is nonzero and the denominator is zero.) Root functions are continuous, except at x-values that would result in a negative value under an even root For piecewise functions, find the f(x) values at the x-value separating the regions of the function. – If the f(x) values are equal, the function is continuous. – Otherwise, there is a (non-removable) discontinuity at this point. 7

Increasing and Decreasing Functions

Definitions Given function f defined on an interval – For any two numbers x 1 and x 2 on the interval Increasing function – f(x 1 ) < f(x 2 ) when x 1 < x 2 Decreasing function – f(x 1 ) > f(x 2 ) when x 1 < x 2 Constant Function – f(x 1 ) = f(x 2 ) when x 1 < x 2 9 X2X2 X1X1 X1X1 X2X2 f(x)

Check These Functions By graphing on calculator, determine the intervals where these functions are – Increasing – Decreasing 10

Notes Over 2.3 Increasing and Decreasing Functions Describe the increasing and decreasing behavior. The function is decreasing on the interval increasing on the interval decreasing on the interval increasing on the interval

Decreasing on(-∞, -1) U (0,1) Increasing on (-1,0) U (1,∞) Using compound Interval Notation is More Effective

Notes Over 2.3 Increasing and Decreasing Functions Describe the increasing and decreasing behavior. The function is increasing on the interval constant on the interval decreasing on the interval

Applications Digitari, the great video game manufacturer determines its cost and revenue functions – C(x) = 4.8x x 2 0 ≤ x ≤ 2250 – R(x) = 8.4x -.002x 2 0 ≤ x ≤ 2250 Determine the interval(s) on which the profit function is increasing 14