Question

How do you determine \[f(x) = {x^4} - {x^{12}} + 1\]is an even or odd function?

Answer 1

Hint: Here the given question is to find the function to be odd or even function, here we need to use the formulae which determines that the function is either even or odd function, the formulae used for determining the odd and even function are as, if the negative of a function and function itself are added together and gives the value as zero then the function is even function and if subtracted and the value is zero then the function is odd function.
Formulae Used:
Even function:
\[f(x) + f( - x) = 0\]
Odd function:
\[f(x) - f( - x) = 0\]

Complete step by step solution:
To solve the given question here we need to use the formulae which determines the function is odd or even, here we will put the expression in the formulae and then check for the function for its even or odd property, the formulae used are:
Even function:
\[f(x) + f( - x) = 0\]
Odd function:
\[f(x) - f( - x) = 0\]
Now for solving the question we need to solve for the function, on putting the value in the formulae we get:
First we are checking for the function is to be odd function we get:
\[
   \Rightarrow {x^4} - {x^{12}} + 1 - [({( - x)^4} - {( - x)^{12}} + 1)] = 0 \\
   \Rightarrow {x^4} - {x^{12}} + 1 - [{x^4} - {x^{12}} + 1] = 0 \\
   \Rightarrow {x^4} - {x^{12}} + 1 - {x^4} + {x^{12}} - 1 = 0 \\
   \Rightarrow 0 = 0 \;
 \]
Hence the given function is an odd function.

Now checking the function to be even function we get:
\[
   \Rightarrow {x^4} - {x^{12}} + 1 + [({( - x)^4} - {( - x)^{12}} + 1)] = 0 \\
   \Rightarrow {x^4} - {x^{12}} + 1 + [{x^4} - {x^{12}} - 1] = 0 \\
   \Rightarrow {x^4} - {x^{12}} + 1 + {x^4} - {x^{12}} - 1 = 0 \\
   \Rightarrow 2{x^4} - 2{x^{12}} = 0\,(L.H.S \ne R.H.S) \;
 \]
Hence the given function is an not an even function.

Note: Here the given question is to check for the function is to be even or odd, here in order to solve the question we can get with the formulae also, and if we can draw the graph for the given function, then function can be checked by plotting the graph of the equation.