Lecture 1. Even / odd functions

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Definition

  • ƒ f with hook is even function: ƒ⁡(-x)=ƒ⁡(x) f with hook of negative x equals f with hook of x


"Symmetric about "y y-axis"

  • cos⁡x cosine x is an even function

  • ƒ f with hook is odd if ƒ⁡(-x)=-ƒ⁡(x) f with hook of negative x equals negative f with hook of x

Odd function: "Symmetric about

the origin"

  • sin⁡x sine x is an odd function

Examples

(a) ƒ⁡(x)=x2+1⁢cos⁡x f with hook of x equals the square root of x squared plus 1 cosine x is even because

ƒ⁡(-x)=(−x)2+1⁢cos⁡(−x) f with hook of negative x equals the square root of open paren negative x close paren squared plus 1 cosine negative x
=x2+1⁢cos⁡x=ƒ⁡(x) equals the square root of x squared plus 1 cosine x equals f with hook of x

(b) g⁡(x)=x3+sin⁡x g of x equals the cube root of x plus sine x is odd because

g⁢(−x)=−x3+sin⁡(−x) g times negative x equals the cube root of negative x plus sine negative x
=−x3−sin⁡x equals negative the cube root of x minus sine x
=−(x3+sin⁡x)=−g⁡(x) equals negative open paren the cube root of x plus sine x close paren equals negative g of x

(c) h⁡(x)=ℇx⁢1+x3 h of x equals ℇ to the x-th power the square root of 1 plus x cubed is neither even or odd because

=ℇ−x⁢1−x3≠h⁡(x) equals ℇ raised to the negative x power the square root of 1 minus x cubed is not equal to h of x or -h⁡(x) negative h of x.

To be continued. Join Rutgers STEM NB community on Corca via this link