This function f(x) = 3x^2 + 2x - 1 is even?

Answer 1

To test if a function is odd/even, you require to know these definitions:

f(x) is even if and only if f(x) = f(-x).
f(x) is odd if and only if f(-x) = -f(x).

Let's test if it is even.

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

f(-x) = 3[-x]^2 + 2[-x] - 1
f(-x) = 3x^2 - 2x - 1, and as you can see, this is not equal to
f(x), so this is not even.

Test for odd: Note that we've already calculate f(-x), so all we have to do is compare it to -f(x).

- f(x) = (-1)f(x) = (-1)(3x^2 + 2x - 1) = -3x^2 - 2x + 1, which is
NOT equal to f(-x).

Therefore the function is neither odd nor even.

Answer 2

No, this function is not.

An even function is one that is symmetric with respect to the y-axis (the part of the graph that's on the left side of the y-axis is the mirror image of the part on the right side of the y-axis). This also means that f(x) = f(-x). So, just test a value of x.
How about x = 1.

f(1) = 3(1)^2 +2(1) - 1 = 4
f(-1) = 3(-1)^2 + 2(-1) - 1 = 0

Since f(1) does not equal f(-1) you know the function is not even.

Answer 3

No f(-x)= 3x^2-2x -1 is not equal f(x) for every x

Answer 4

No it's not. You don't need any long-winded tests. The presence of the 2x in the middle automatically means that it can't be an even function as these are ones that only have even powers of x in the function or its McLaurin expansion.

Answer 5

Back in elementary school you learned the following:

even * even = even
even * odd = even
odd * odd = old

Let's break apart your original equation: f(x) = 3x^2 + 2x - 1.

2x will always be even.

3x^2 could be rewritten 3 * x^2
x^2 will be even for even values, odd for odd values of x.
The resulting even or odd value is then multipled by the odd number 3. The result remains even for even values of x and remains odd for odd values of x.


Back in school we also learned:

even - even = even (ex. 6 - 2 = 4)
even - odd = odd (ex. 8 - 3 = 5)
odd - odd = even (ex. 7 - 3 = 4)
odd - even = odd (ex. 9 - 4 = 5)

Subtracting one (or any odd number) from an even value gives you an odd number.
Subtracting one (or any odd value) from an even number gives you an odd value.


Putting this together our function becomes:

f(even x) = even - even - 1 = odd
f(odd x) = odd - even - 1 = even

Answer 6

If x is an integer, then 2x-1 is always odd.

However, 3x^2 will be even when, and only when x is even. SO....

f(x even) = odd and
f(x odd)= even

Answer 7

3x^2+2x-1 = (3x-1)(x+1)
So for f(x) to be even, either 3x-1 or x+1 must be even
So f(x) is even for all odd numbers

Answer 8

No, it's not even. A polynomial function is even (in uiversal domain) only if the coefficients of all odd powers of x are zero.