Question 138650


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^4-1}}} is an even function.



{{{f(x)=x^4-1}}} Start with the given function.



{{{f(-x)=(-x)^4-1}}} Replace each x with -x.



{{{f(-x)=x^4-1}}} Simplify. Note: only the terms with an <b>odd</b> exponent will change in sign.


So this shows us that {{{x^4-1=x^4-1}}} which means that {{{f(x)=f(-x)}}}

Since {{{f(x)=f(-x)}}}, this shows us that {{{f(x)=x^4-1}}} is an even function.



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Now, let's see if {{{f(x)=x^4-1}}} is an odd function.


{{{f(x)=x^4-1}}} Start with the given function.



{{{-f(x)=-(x^4-1)}}} Negate the entire function by placing a negative outside the function.



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



So this shows us that {{{x^4-1<>-x^4}}} which means that {{{f(-x)=-f(x)}}}

Since {{{f(-x)<>-f(x)}}}, this shows us that {{{f(x)=x^4-1}}} is <b>not</b> an odd function.



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Answer:

So the function {{{f(x)=x^4-1}}} is an even function.