Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 163.3 Properties of FunctionsChapter 3 Section 3: Properties of FunctionsIn this section, we will…Determine if a given function is even, odd or neitherDetermine where a function is increasing, decreasing or constantFind local maxima and minima3.3 Determine if a Function is Even, Odd or NeitherA function is even if, for every number in the domain, is also in the domain and ( ) ( )f xx f x f x- - =If you replace x with –x in the equation and an equivalent equation results, the function is even.What symmetry is this?3.3 Determine if a Function is Even, Odd or NeitherIf you replace x with –x in the equation and an equation which has one side negated results, the function is odd.A function is odd if, for every number in the domain, is also in the domain and ( ) ( )f xx f x f x- - =-What symmetry is this?Example: Determine graphically whether each function given is an even function, an odd function, or a function that is neither even nor odd.3.3 Determine if a Function is Even, Odd or NeitherExample: Determine algebraically whether each function given is an even function, an odd function, or a function that is neither even nor odd.3.3 Determine if a Function is Even, Odd or Neither3( ) 4 5f x x x= -3( ) 4 5 7g x x x= - +4 2( ) 3h x x x=- +4 2( ) 3 7f x x x=- + +3.3 Determine if a Function is Even, Odd or Neither3( )f x x=2( )g x x x= +81( )h xx=32( )3 9xf xx-=-Example: Determine algebraically whether each function given is an even function, an odd function, or a function that is neither even nor odd.3.3 Determine where a Function is Increasing, Decreasing or Neither( ) ( )1 2 1 21 2A function is increasing over an open interval , if for any choice of and in , with we have .f Ix x I x xf x f x<<Where is the function increasing?We talk about a function increasing over an open interval of x-values.( ) ( )1 2 1 21 2A function is decreasing over an open interval , if for any choice of and in , with we have .f Ix x I x xf x f x<>Where is the function decreasing?We talk about a function decreasing over an open interval of x-values.3.3 Determine where a Function is Increasing, Decreasing or Neither( ) ( )1 2 1 21 2A function is constant over an open interval , if for any choice of and in , with we have .f Ix x I x xf x f x<=Where is the function constant?We talk about a function being constant over an open interval of x-values.3.3 Determine where a Function is Increasing, Decreasing or NeitherExample: Find the intervals over which the function below is increasing, decreasing or constant.increasing:decreasing: constant:3.3 Determine where a Function is Increasing, Decreasing or NeitherExample: Find the intervals over which the function is increasing, decreasing or constant; consider the function over (-2, 5).increasing:decreasing:constant:3 2( ) 3 5f x x x= - +What if we considered the function over its entire domain?3.3 Determine where a Function is Increasing, Decreasing or NeitherA function has a local maximum at if there is an open interval containing so that, for all not equal to in , ( ) ( ).We call ( ) a local maximum of .f cI cx c I f x f cf c f<3.3 Finding Local Maxima and MinimaWhat are the local maxima?At what values do the local maxima occur?A function has a local minimum at if there is an open interval containing so that, for all not equal to in , ( ) ( ).We call ( ) a local minimum of .f cI cx c I f x f cf c f>3.3 Finding Local Maxima and MinimaWhat are the local minima?At what values do the local minima occur?Example: Find the local maxima and local minima of the function; determine where the function is increasing, where it is decreasing and where it is constant:Round answers to the nearest hundredth.local minimum: increasing:local maximum: decreasing:constant:3.3 Properties of Functions4 3( ) 0.8 2 1.f x x x x= - + +Example: The height h of a ball (in feet) thrown with an initial velocity of 80 feet per second from an initial height of 6 feet is given as a function of the time t (in seconds) by• Graph h• Determine the time at which the height is at its maximum.• What is the maximum height?• State the intervals over which the function is increasing and decreasing. Describe what this means in the context of the problem.• increasing:• decreasing:• interpretation:3.3 Properties of Functions2( ) 16 80 6.h t t t=- + +Independent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect.Read pages 231-238 Homework: pp. 238-241#21-49 odds, 653.3 Properties of