EVEN AND ODD FUNCTIONS

• Recall that $\,x\,$ and $\,-x\,$ are opposites.
  When $\,x\,$ is the input, $\,f(x)\,$ is the output.
  When $\,-x\,$ is the input, $\,f(-x)\,$ is the output.
  For even functions, these two outputs must be equal.
• Note: A function $\,f\,$ is even if and only if the graph of $\,f\,$ is symmetric about the $\,y\,$-axis.
• Note:   $\,y = x^2\,$,   $\,y = x^4\,$,   $\,y = x^6\,$,   $\,y = x^8\,$, etc., are all even functions.
  Thus, power functions with even powers are even functions!
  So, the name ‘even’ seems reasonable.
• There are lots of other even functions.
  For example, $\,y = |x|\,$ and $\,y = \cos(x)\,$ are even functions.
• Is   $\,g(t) = t^2 - t^4\,$   an even function?
  You must investigate $\,g(-t)\,$, and compare it with $\,g(t)\,$, as follows:
  
  

  $g(-t)$	$=$	$(-t)^2 - (-t)^4$	definition of $\,g\,$; function evaluation
  	$=$	$t^2 - t^4$	simplify
  	$=$	$g(t)$	definition of $\,g$

  
  Thus, $\,g\,$ is an even function.

ODD FUNCTIONS

for odd functions:
when inputs are opposites, the corresponding outputs are opposites

• Recall that $\,x\,$ and $\,-x\,$ are opposites.
  When $\,x\,$ is the input, $\,f(x)\,$ is the output.
  When $\,-x\,$ is the input, $\,f(-x)\,$ is the output.
  For odd functions, these two outputs must be opposites.
• Observe that ‘$\,f(x) = -f(-x)\,$’ is equivalent to ‘$\,f(-x) = -f(x)\,$’ (multiply both sides by $\,-1\,$ and switch sides).
  Sometimes, the definition is given using this equivalent characterization.
• Note: A function $\,f\,$ is odd if and only if the graph of $\,f\,$ is symmetric about the origin.
• Note:   $\,y = x\,$,   $\,y = x^3\,$,   $\,y = x^5\,$,   $\,y = x^7\,$,   $\,y = x^9\,$, etc., are all odd functions.
  Thus, power functions with odd powers are odd functions!
  So, the name ‘odd’ seems reasonable.
• There are lots of other odd functions.
  For example, $\displaystyle \,y = \frac{1}{x}\,$ and $\,y = \sin(x)\,$ are odd functions.
• Is   $\,g(x) = x^3 - 1\,$   an odd function?
  That is, are $\,g(-x)\,$ and $\,g(x)\,$ opposites?
  That is, are $\,g(-x)\,$ and $\,-g(x)\,$ equal?
  
  

  $\cssId{s72}{g(-x)} \cssId{s73}{= (-x)^3 - 1} \cssId{s74}{= -x^3 - 1}$	find $\,g(-x)\,$:   definition of $\,g\,$;   function evaluation
  $\cssId{s77}{-g(x)} \cssId{s78}{= -(x^3-1)} \cssId{s79}{= -x^3 + 1}$	find the opposite of $\,g(x)\,$:   definition of $\,g\,$;   the distributive law

  
  Since $\,g(-x)\,$ and $\,-g(x)\,$ are different, $\,g\,$ is not an odd function.
• If an odd function is defined at zero, then its graph must pass through the origin.
  Why?
  If $\,f\,$ is odd and defined at zero, then:
  
  

  $f(x) = -f(-x)$	requirement for an odd function
  $f(0) = -f(-0)$	let $\,x = 0$
  $f(0) = -f(0)$	the opposite of zero is zero
  $2f(0) = 0$	add $\,f(0)\,$ to both sides
  $f(0) = 0$	divide both sides by $\,2$