even and odd functions

​Odd Functions

​A function is said to be odd if its graph is symmetric with respect to the origin. f(-x)=-f(x)

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The following graph is even, because is you fold it over the y-axis, the left and right sides will match​

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The following graph is odd, because is you fold it over the origin, both parts will match​

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The following graph is neither even nor odd because if you fold it over the origin or the y-axis the points will not match

​Ex: Decide whether the function is even, odd or neither

g(x)=x³-x

Solution: Replace x for -x

g(-x)= (-x​)³-(-x)= -x³+x

Now compare the signs of all terms in the new function with the original function, if all signs change, is an ODD function​

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​Ex: Decide whether the function is even, odd or neither

g(x)= x⁴+x²+1

Solution: Replace x for -x

g(-x)= (-x​)⁴+ (-x)² + 1= x⁴+x²+1

Now compare the signs of all terms in the new function with the original function, if all signs remain the same, is an EVEN function​

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​Ex: Decide whether the function is even, odd or neither

g(x)= x⁴+x³

Solution: Replace x for -x

g(-x)= (-x​)⁴+ (-x)³ = x⁴-x³

Now compare the signs of all terms in the new function with the original function, if only some signs change, NEITHER

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