Tuesday, August 15, 2017

High School Solutions – Functions Calculator, Parity (Even or Odd)

In this blog post, we will be discussing the parity of functions and how to find out the parity of a function.

The parity of a function is the attribute of being even, odd, or neither. A function is even, if \f(-x)=\f(x) for all x. A function is odd if \f(-x)=-\f(x) for all x. A function is neither even nor odd, when it satisfies neither of these options.

In order to find the parity of a function, we must see whether the statements about even and odd functions is true or false.

Steps to find the parity of a function:
  1.  Find \f(-x)
  2.  Find -\f(x)
  3.  See if \f(-x)=\f(x), \f(-x)=-\f(x), or neither
Simple enough! Let’s move onto some examples.

First example (click here):

                                                 Find the parity of \f(x)=x^2+4

1. Find \f(-x)

                                                         \f(-x)=(-x)^2+4

                                                               \f(-x)=x^2+4

2. Find -\f(x)

                                                      -\f(x)=-(x^2+4)=-x^2-4

3.  See if \f(-x)=\f(x),\f(-x)=-\f(x), or neither

                                                         \f(x)=x^2+4=\f(-x)

                                                  \f(-x)=x^2+4≠-x^2-4=-\f(x)

             \f(x) is even.

Next example (click here):

                                                   Find the parity of \f(x)=3x

1. Find f(-x)

                                                          \f(-x)=3(-x)=-3x

2. Find -\f(x)

                                                          -\f(x)=-(3x)=-3x

3. See if \f(-x)=\f(x),\f(-x)=-\f(x), or neither

                                                        \f(-x)=-3x≠3x=\f(x)

                                                          \f(-x)=-3x=-\f(x)

\f(x) is odd.

Last example (click here):

                                             Find the parity of \f(x)=\frac{x+2}{x+1}

1. Find \f(-x)

                                                        \f(-x)=\frac{-x+2}{-x+1}

2. Find -\f(x)

                                             -\f(x)=-(\frac{x+2}{x+1})=\frac{-x-2}{-x-1}

3. See if \f(-x)=\f(x),\f(-x)=-\f(x), or neither

                                          \f(-x)=\frac{-x+2}{-x+1}≠\frac{x+2}{x+1}=\f(x)

                                         \f(-x)=\frac{-x+2}{-x+1}≠\frac{-x-2}{-x-1}=-\f(x)

\f(x) is neither odd nor even.

As you can see, finding the parity of a function is very simple. Just memorize the definitions of even and odd functions and you are good to go! For more help or practice on this topic, check out Symbolab’s Practice.

Until next time,

Leah

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