SOLUTION: Find if the function is even, odd or neither. f(x)=x^4-1

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Question 138650: Find if the function is even, odd or neither.

f(x)=x^4-1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Remember, if f%28x%29=f%28-x%29 then the function is an even function. If f%28-x%29=-f%28x%29 then the function is an odd function.



First, let's see if f%28x%29=x%5E4-1 is an even function.


f%28x%29=x%5E4-1 Start with the given function.


f%28-x%29=%28-x%29%5E4-1 Replace each x with -x.


f%28-x%29=x%5E4-1 Simplify. Note: only the terms with an odd exponent will change in sign.

So this shows us that x%5E4-1=x%5E4-1 which means that f%28x%29=f%28-x%29
Since f%28x%29=f%28-x%29, this shows us that f%28x%29=x%5E4-1 is an even function.


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Now, let's see if f%28x%29=x%5E4-1 is an odd function.

f%28x%29=x%5E4-1 Start with the given function.


-f%28x%29=-%28x%5E4-1%29 Negate the entire function by placing a negative outside the function.


-f%28x%29=-x%5E4 Distribute and simplify.


So this shows us that x%5E4-1%3C%3E-x%5E4 which means that f%28-x%29=-f%28x%29
Since f%28-x%29%3C%3E-f%28x%29, this shows us that f%28x%29=x%5E4-1 is not an odd function.


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Answer:
So the function f%28x%29=x%5E4-1 is an even function.