How to Tell if a Function Is Even or Odd

Step-by-Step Guide

1. 1

   Step 1: Review opposite variables.

   In algebra, the opposite of a variable is written as a negative.
   
   This is true whether the variable in the function is x{\displaystyle x} or anything else.
   
   If the variable in the original function already appears as a negative (or a subtraction), then its opposite will be a positive (or addition).
   
   The following are examples of some variables and their opposites:the opposite of x{\displaystyle x} is −x{\displaystyle
   -x} the opposite of q{\displaystyle q} is −q{\displaystyle
   -q} the opposite of −w{\displaystyle
   -w} is w{\displaystyle w}.
2. 2

   Step 2: Replace each variable in the function with its opposite.

   Do not alter the original function other than the sign of the variable.
   
   For example:f(x)=4x2−7{\displaystyle f(x)=4x^{2}-7} becomes f(−x)=4(−x)2−7{\displaystyle f(-x)=4(-x)^{2}-7} g(x)=5x5−2x{\displaystyle g(x)=5x^{5}-2x} becomes g(−x)=5(−x)5−2(−x){\displaystyle g(-x)=5(-x)^{5}-2(-x)} h(x)=7x2+5x+3{\displaystyle h(x)=7x^{2}+5x+3} becomes h(−x)=7(−x)2+5(−x)+3{\displaystyle h(-x)=7(-x)^{2}+5(-x)+3}. , At this stage, you are not concerned with solving the function for any particular numerical value.
   
   You simply want to simplify the variables to compare the new function, f(-x), with the original function, f(x).
   
   Remember the basic rules of exponents which say that a negative base raised to an even power will be positive, while a negative base raised to an odd power will be negative.f(−x)=4(−x)2−7{\displaystyle f(-x)=4(-x)^{2}-7} f(−x)=4x2−7{\displaystyle f(-x)=4x^{2}-7} g(−x)=5(−x)5−2(−x){\displaystyle g(-x)=5(-x)^{5}-2(-x)} g(−x)=5(−x5)+2x{\displaystyle g(-x)=5(-x^{5})+2x} g(−x)=−5x5+2x{\displaystyle g(-x)=-5x^{5}+2x} h(−x)=7(−x)2+5(−x)+3{\displaystyle h(-x)=7(-x)^{2}+5(-x)+3} h(−x)=7x2−5x+3{\displaystyle h(-x)=7x^{2}-5x+3} , For each example that you are testing, compare the simplified version of f(-x) with the original f(x).
   
   Line up the terms with each other for easy comparison, and compare the signs of all terms.If the two results are the same, then f(x)=f(-x), and the original function is even.
   
   An example is: f(x)=4x2−7{\displaystyle f(x)=4x^{2}-7} and f(−x)=4x2−7{\displaystyle f(-x)=4x^{2}-7}.
   
   These two are the same, so the function is even.
   
   If each term in the new version of the function is the opposite of the corresponding term of the original, then f(x)=-f(-x), and the function is odd.
   
   For example: g(x)=5x5−2x{\displaystyle g(x)=5x^{5}-2x} but g(−x)=−5x5+2x{\displaystyle g(-x)=-5x^{5}+2x}.
   
   Notice that if you multiply each term of the first function by
   -1, you will create the second function.
   
   Thus, the original function g(x) is odd.
   
   If the new function does not meet either of these two examples, then it is neither even nor odd.
   
   For example: h(x)=7x2+5x+3{\displaystyle h(x)=7x^{2}+5x+3} but h(−x)=7x2−5x+3{\displaystyle h(-x)=7x^{2}-5x+3}.
   
   The first term is the same in each function, but the second term is an opposite.
   
   Therefore, this function is neither even nor odd.
3. 3

   Step 3: Simplify the new function.

4. 4

   Step 4: Compare the two functions.

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