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