Question 200147


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)=5x^3-9}}} is an even function.



{{{f(x)=5x^3-9}}} Start with the given function.



{{{f(-x)=5(-x)^3-9}}} Replace each x with -x.



{{{f(-x)=-5x^3-9}}} Simplify. Note: only the terms with an <b>odd</b> exponent will change in sign.


So this shows us that {{{5x^3-9<>-5x^3-9}}} which means that {{{f(x)<>f(-x)}}}


Since {{{f(x)<>f(-x)}}}, this shows us that {{{f(x)=5x^3-9}}} is <b>not</b> an even function.



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Now, let's see if {{{f(x)=5x^3-9}}} is an odd function.


{{{f(x)=5x^3-9}}} Start with the given function.



{{{-f(x)=-(5x^3-9)}}} Negate the entire function by placing a negative outside the function.



{{{-f(x)=-5x^3+9}}} Distribute and simplify.



So this shows us that {{{-5x^3-9<>-5x^3+9}}} which means that {{{f(-x)<>-f(x)}}}


Since {{{f(-x)<>-f(x)}}}, this shows us that {{{f(x)=5x^3-9}}} is <b>not</b> an odd function.




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Answer:


So the function {{{f(x)=5x^3-9}}} is neither an even nor odd function.