Determine algebraically whether the function is even, odd, or neither.  Give reasons for your answer. f(x)=(3x+2)^2 (x-4)(x+1)(2x-3)

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beckden | High School Teacher | (Level 1) Educator

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`f(x) = (3x+2)^2(x-4)(x+1)(2x-3)`

A function is even if `f(-x) = f(x)` for all x   An example would be f(x)=x^4

A function is odd if `f(-x) = -f(x)` for all x   An example would be f(x)=x^3

If neither one of these is true then the function is neither even nor odd.  An example would be f(x) = 2x + 1

So we want to calculate f(-x)

`f(-x) = (-3x+2)^2 (-x-4)(-x+1)(-2x-3)` Now we simplify

`f(-x) = ((-1)(3x - 2))^2 ((-1)(x+4))((-1)(x-1))((-1)(2x+3)`

`f(-x) = (-1)^2 (-1)(-1)(-1) (3x - 2)^2 (x+4)(x-1)(2x+3)`

`f(-x) = -(3x-2)^2 (x+4)(x-1)(2x+3)`

We can see that f(-x) `!=` f(x) and f(-x) `!=` -f(x)

So the function is neither even or odd.

 

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