An function f(x)f(x) is even if f("-"x)=f(x)f(-x)=f(x). What does this mean? It means that for any given input xx, the function ff does the same thing to xx as it does to -x−x. When we give it xx, it gives us yy back. When we give it -x−x, it gives us yy again.
The visual interpretation to this is that an even function is symmetrical about the yy-axis. It has to be, because no matter what xx we choose, the yy-value for xx matches the yy-value for -x−x.
To see if a function ff is even, we ask: when ff gets the input -x−x, does it return the same output as if we had given it xx? In math terms, we're asking:
Does f("-"x)=f(x)f(-x)=f(x)?
For the given function f(x)=x^4-64f(x)=x4−64, ff takes an input, computes its 4th power, then subtracts 64. To test if ff is even, we plug -x−x in as our input and see if we still get x^4-64x4−64 back.
We compute:
f("-"x)=("-"x)^4 -64f(-x)=(-x)4−64
color(white)(f("-"x))=("-"1)^4(x)^4 -64f(-x)=(-1)4(x)4−64
color(white)(f("-"x))=1xx x^4 -64f(-x)=1×x4−64
color(white)(f("-"x))=x^4 -64f(-x)=x4−64
And hey look: f("-"x)=x^4-64f(-x)=x4−64, which is f(x)f(x)! Thus, f("-"x)=f(x)f(-x)=f(x), and so ff is even.
