Even and odd functions

Even and odd functions

Even functions

• f(x) = f(-x) for all x

• there is symmetry (a reflection) over the y-axis

• They got called "even" functions because the functions x², x⁴, x⁶, x⁸, etc behave like that, but there are other functions that behave like that too.

Don't just look at the exponent

• But an even exponent does not always make an even function, for example (x+1)² is not an even function.

• There is not symmetry over the y-axis

• 
  

Odd functions

• −f(x) = f(−x) for all x

• this is called origin symmetry

• They got called "odd" because the functions x, x³, x⁵, x⁷, etc behave like that, but there are other functions that behave like that, too

• But an odd exponent does not always make an odd function

• There is no symmetry about the origin

• AND

   $-f\left(x\right)\ne f\left(-x\right)$  

Neither even or odd

• Don't be misled by the names "odd" and "even" ... they are just names ... and a function does not have to be even or odd.

• In fact most functions are neither odd nor even.

Ex. 1

• Is f even?

• If f odd?

• Neither?

Ex. 1 is f even?

• You could check the symmetry over the y axis.

• Or you could check points.

• For even functions f(x) = f(-x)

• when x = 1, f(x) = 2

• when x = -1, f(x) = 2

• so f(x) = f(-x)

• this is true for every point on f soo the function is even

Ex. 1 is f odd?

• for odd functions f(-x) = -f(x)

• (-1, 2) means f(-1) = 2

• but -f(1) = -2

•  $f\left(-x\right)\ne f\left(x\right)$  

• f is not odd

Multiple Choice

Is f even?

• yes

• no

• can't tell

f is not even

• There is not symmetry over the y axis

• f(1) = 4

• f(-1) = 2

•  $f\left(x\right)\ne f\left(-x\right)$  

Multiple Choice

if f odd?

• yes

• no

• can't tell

f is NOT odd

• in an odd function f(-x) = -f(x)

• f(-1) = 2

• but -f(1) = -4

Ex. 3

• Is f even, odd, or neither?

• Lets check even first. For even functions f(x) = f(-x)

• f(6) = -4

• f(-6) = -8

•  $f\left(x\right)\ne f\left(-x\right)$  

• NOT even

Ex. 3

• Is f even, odd, or neither?

• Lets check odd.  For odd functions -f(x) = f(-x)

• f(-6) = -8

• for f(-x), f(6) = -4

• for -f(x), -f(-6) = 8

•  $f\left(-x\right)\ne-f\left(x\right)$  so its not odd
  

• Its neither

Multiple Choice

Ex. 4 is f even?

• yes

• no

• For even functions f(x) = f(-x)

• f(-2) = -3

• f(2) = 3

•  $f\left(x\right)\ne f\left(-x\right)$  

• NOT EVEN

Multiple Choice

Ex. 4 is f odd?

• yes

• no

• for odd functions -f(x) = f(-x)

• f(-2) = -3

• for -f(x): -f(-2) = 3

• for f(-x): f(2) = 3

• -f(x) = f(-x)

• it is odd

How to check if an equation is even, odd, or neither

• Step 1: find f(-x)

• Step 2: find -f(x)

• Step 3 compare

• if f(x) = f(-x), function is even

• if -f(x) = f(-x), function is odd

• or neither

ex. 5 is

 $f\left(x\right)=x^3-1\$  even, odd, or neither

•  $f\left(-x\right)=\left(-x\right)^3-1$  

•  $f\left(-x\right)=-x^3-1$  

•  $-f\left(x\right)=-\left(x^3-1\right)$  

•  $-f\left(x\right)=-x^3+1$  

•  $f\left(x\right)\ne f\left(-x\right)$  

•  $-f\left(x\right)\ne f\left(-x\right)$  

• neither

ex. 6 is

 $f\left(x\right)=2^{x+3^{ }}$  even, odd, or neither

•  $f\left(-x\right)=2^{-x+3^{ }}$  

•  $-f\left(x\right)=-\left(2^{x+3^{ }}\right)$  

•  $-f\left(x\right)=-2^{x+3^{ }}$  

•  $f\left(x\right)\ne f\left(-x\right)$  not even

•  $-f\left(x\right)\ne f\left(-x\right)$  not odd

• neither

Multiple Choice

ex. 7 is

 $f\left(x\right)=x^4-2x^2$  even, odd, or neither?

• even

• odd

• neither

 $f\left(x\right)=x^4-2x^2$  

•  $f\left(-x\right)=\left(-x\right)^4-2\left(-x\right)^2$  

•  $f\left(-x\right)=x^4-2x^2$  

• EVEN

Homework in Khan Academy

• 1. Even and odd functions graphs and tables

• 2. Even and odd functions equations

• 3. Transformations of functions quiz 1