Section 2.3 Properties of Functionsfaculty.montgomerycollege.edu/maronne/Ma180/power... · 2 −3. gx x (−=−) 2 ( ) 2 −3 = −=2 3 x fx. 2. Even function symmetric with respect

Section 2.3

Properties of Functions

Copyright © 2013 Pearson Education, Inc. All rights reserved


For an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.


So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.



Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.

Even function because it is symmetric with respect to the y-axis

Neither even nor odd because no symmetry with respect to the y-axis or the origin.

Odd function because it is symmetric with respect to the origin.



( ) 3) 5a f x x x= + ( ) ( ) ( )3 5f x x x− = − + − 3 5x x= − −

( ) ( )3 5x x f x= − + = −Odd function symmetric with respect to the origin

( ) 2) 2 3b g x x= − ( ) ( )2 32g x x− = − − 22 3 ( )x f x= − =

Even function symmetric with respect to the y-axis

( ) 3) 14c h x x= − + ( ) ( )34 1h x x− = − − + 34 1x= +Since the resulting function does not equal f(x) nor –f(x) this function is neither even nor odd and is not symmetric with respect to the y-axis or the origin.



Where is the function increasing?


Where is the function decreasing?


Where is the function constant?








There is a local maximum when x = 1.

The local maximum value is 2


There is a local minimum when x = –1 and x = 3.

The local minima values are 1 and 0. Copyright © 2013 Pearson Education, Inc. All rights reserved

(e) List the intervals on which f is increasing.

(f) List the intervals on which f is decreasing.

( ) ( )1,1 and 3,− ∞

( ) ( ), 1 and 1,3−∞ −




Find the absolute maximum and the absolute minimum, if they exist.

The absolute maximum of 6 occurs when x = 3.

The absolute minimum of 1 occurs when x = 0.




There is no absolute minimum because of the “hole” at x = 3.




The absolute minimum of 1 occurs on the interval [1,2].



There is no absolute maximum.

The absolute minimum of 0 occurs when x = 0.



There is no absolute maximum.

There is no absolute minimum.




( ) 3Use a graphing utility to graph 2 3 1 for 2 2.Approximate where has any local maxima or local minima.

f x x x xf

= − + − < <


( ) 3Use a graphing utility to graph 2 3 1 for 2 2.Determine where is increasing and where it is decreasing.

f x x x xf

= − + − < <




a) From 1 to 3


b) From 1 to 5


c) From 1 to 7




( ) 2Suppose that 2 4 3.g x x x= − + −

( ) ( )( )( )

222(1) 4(1) 3 2 2 4 2 3 18(a) 61 2 3

yx

− + − − − − + − −∆= = =

∆ − −

( ) ( 19) 6( ( 2))b y x− − = − −

19 6 12y x+ = +

6 7y x= −

-4 3

2

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Section 2.3 Properties of Functionsfaculty.montgomerycollege.edu/maronne/Ma180/power... · 2 −3. gx x (−=−) 2 ( ) 2 −3 = −=2 3 x fx. 2. Even function symmetric with respect