Is the function `f(x) = 13x^4 – 2x^3 + 7x` even, odd or neither?

Perform a test:

For even functions:
`f(x)= f(-x)`

For odd functions:
`f(x)= -f(x)`

Negate the `x` s in the equation and let's see:
`f(x) = 13(-x)^4 – 2(-x)^3 + 7(-x)`

`f(x) = 13(x)^4 +2(x)^3 - 7(x)`

Is this equal to:
`f(x)`, if not this is not an even function

`-f(x)`, if not this is not an odd function
`-f(x)= -13(x)^4 + 2(x)^3 -7(x)`

Since it is equal to neither, this function is neither odd or even

Given: `f(x) = 14x^4 - 2x^3 +7x`

If a function is even, it is symmetric about the `y-` axis: `f(-x) = f(x)`

If a function is odd, it is symmetric about origin `y-` axis: `f(-x) = -f(x)`

Assuming we don't have access to a graphing calculator, let's test for even #f(-x) = f(x):

original function: `f(x) = 14x^4 - 2x^3 +7x`

Replace `x` by `-x`:
`f(-x) = 14(-x)^4 - 2(-x)^3 +7(-x)`

`f(-x) = 14x^4 + 2x^3 -7x != f(x)" "` Not an even function

`--------------------`
Test for an odd function, `f(-x) = -f(x)`:

original function: `f(x) = 14x^4 - 2x^3 +7x`

`-f(x) = -(14x^4 - 2x^3 +7x) = -14x^4 + 2x^3 -7x`

Replace `x` by `-x` and `f(x)` by `-f(x)`:
`-f(-x) = 14(-x)^4 - 2(-x)^3 +7(-x)`

`f(-x) = -(14x^4 + 2x^3 -7x)`

`f(-x) = -14x^4 - 2x^3 +7x != - f(x)`