Question

Is the function `y=x^4` odd, even, or neither?

Answer 1

It is even. See explanation.

Explanation:

To check if a function is even, odd or neither you have to calculate `f(-x)` and compare the result with the original function:

• if `f(-x)=f(x)` then the function is even,

• if `f(-x)=-f(x)` then the function is odd,

• in other cases the function is neither odd or even.

Here we have:

`f(x)=x^4`

`f(-x)=(-x)^4=x^4`

`f(-x)=f(x)`, so the function is even.

Note:

Generally the function `y=x^n` is odd for odd values of the exponent and even for even exponent.

Other way of checking if the function is odd or even is to look at its graph:

The `Y` axis is the axis of symetry of an even function. The example can be:

graph{x^2 [-10, 10, -5, 5]}

If a function is even then the origin is its center of symetry:

graph{x^5 [-10,10,-5,5]}

Answer 2

`y=x^4" is even"`

Explanation:

`"to determine if y is odd/even"`

`• " if "f(x)=f(-x)" then "f(x)" is even"`

`"even functions are symmetrical about the y-axis"`

`• " if "f(-x)=-f(x)" then "f(x)" is odd"`

`"odd functions have half turn symmetry about the origin"`

`f(-x)=(-x)^4=x^4=f(x)`

`rArry=x^4" is even"`
graph{x^4 [-10, 10, -5, 5]}