Question 138649


Remember, if {{{f(x)=f(-x)}}} then the function is an even function. If {{{f(-x)=-f(x)}}} then the function is an odd function.




First, let's see if {{{f(x)=x^3+4x}}} is an even function.



{{{f(x)=x^3+4x}}} Start with the given function.



{{{f(-x)=(-x)^3+4(-x)}}} Replace each x with -x.



{{{f(-x)=-x^3-4x}}} Simplify. Note: only the terms with an <b>odd</b> exponent will change in sign.




Since {{{f(x)<>f(-x)}}}, this shows us that {{{f(x)=x^3+4x}}} is <b>not</b> an even function.



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Now, let's see if {{{f(x)=x^3+4x}}} is an odd function.


{{{f(x)=x^3+4x}}} Start with the given function.



{{{-f(x)=-(x^3+4x)}}} Negate the entire function by placing a negative outside the function.



{{{-f(x)=-x^3-4x}}} Distribute and simplify.





Since f(-x)=-f(x), this shows us that {{{f(x)=x^3+4x}}} is an odd function.



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Answer:

So the function {{{f(x)=x^3+4x}}} is an odd function.