SOLUTION: determine whether the function f(x)=3x^3-5x is even, odd, or neither. b) name the graph family for this function.

Algebra ->  Functions -> SOLUTION: determine whether the function f(x)=3x^3-5x is even, odd, or neither. b) name the graph family for this function.      Log On


   

Question 189647: determine whether the function f(x)=3x^3-5x is even, odd, or neither. b) name the graph family for this function.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)


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=3x%5E3-5x is an even function.


f%28x%29=3x%5E3-5x Start with the given function.


f%28-x%29=3%28-x%29%5E3-5%28-x%29 Replace each x with -x.


f%28-x%29=-3x%5E3%2B5x Simplify. Note: only the terms with an odd exponent will change in sign.

So this shows us that 3x%5E3-5x%3C%3E-3x%5E3%2B5x which means that f%28x%29%3C%3Ef%28-x%29


Since f%28x%29%3C%3Ef%28-x%29, this shows us that f%28x%29=3x%5E3-5x is not an even function.


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Now, let's see if f%28x%29=3x%5E3-5x is an odd function.

f%28x%29=3x%5E3-5x Start with the given function.


-f%28x%29=-%283x%5E3-5x%29 Negate the entire function by placing a negative outside the function.


-f%28x%29=-3x%5E3%2B5x Distribute and simplify.


So this shows us that -3x%5E3%2B5x=-3x%5E3%2B5x which means that f%28-x%29=-f%28x%29


Since f%28-x%29=-f%28x%29, this shows us that f%28x%29=3x%5E3-5x is an odd function.


Note: if you graph f%28x%29=3x%5E3-5x, you'll find that the graph has symmetry about the origin.


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Answer:


So the function f%28x%29=3x%5E3-5x is an odd function.


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b) Since the degree of f%28x%29=3x%5E3-5x is 3, this means that this graph is part of a family of cubics.