Question 189647
a)



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



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



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



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


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



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



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


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



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



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



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



Since {{{f(-x)=-f(x)}}}, this shows us that {{{f(x)=3x^3-5x}}} is an odd function.



Note: if you graph {{{f(x)=3x^3-5x}}}, you'll find that the graph has symmetry about the origin.



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Answer:



So the function {{{f(x)=3x^3-5x}}} is an odd function.



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b) Since the degree of {{{f(x)=3x^3-5x}}} is 3, this means that this graph is part of a family of <a href="http://en.wikipedia.org/wiki/Cubic_equation">cubics</a>.