We have to find if f(x) = 3x^4 - 5x^2 + 2 is even or odd.
Now for even function we know that f(-x) = f(x).
Here for f(x) = 3x^4 - 5x^2 + 2 we see that
f(-x) = 3*(-x)^4 - 5(-x)^2 + 2
as for any number raised x raised to an even power n, x^n = (-x)^n
=> 3*x^4 - 5x^2 + 2
Therefore as the function has only even powers of x we can see that it is an even function.
A function is even if
f(-x) = f(x)
In other words, plugging in a number will be the same as plugging in the negative value of the same number.
We'll analyze the given function, replacing each x by -x.
f(-x) = 3(-x)^4 - 5(-x)^2 + 2
We'll compute raising -x to the 4th and 2nd powers and we'll get:
(-x)^4 = (-x)(-x)(-x)(-x) = x^2*x^2 = x^4
f(-x) = 3(x)^4 - 5(x)^2 + 2
So we can see that:
f(-x) = f(x) which means that the function f(x) is an even function.
An even function has symmetry across the y-axis. We also know that if all of the exponents are even, then the function is even.