Determine if Odd, Even, or Neither f(x)=x/(x^2+1)

Problem

ƒ(x)=x/(x2+1)

Solution

1. Substitute −x for every x in the function to find ƒ*(−x)

ƒ*(−x)=(−x)/((−x)2+1)

2. Simplify the expression by squaring the negative variable in the denominator.

ƒ*(−x)=(−x)/(x2+1)

3. Factor out the negative sign from the numerator to compare the result to the original function ƒ(x)

ƒ*(−x)=−x/(x2+1)

4. Identify the relationship between ƒ*(−x) and ƒ(x) Since ƒ*(−x)=−ƒ(x) the function satisfies the definition of an odd function.

−ƒ(x)=−(x/(x2+1))

ƒ*(−x)=−ƒ(x)

Final Answer

ƒ(x)=x/(x2+1)* is Odd

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