Question

How do you determine if $ f\left( x \right) = 3{x^5} - 4x + 3 $ is an even or odd function?

Answer 1

Hint: To determine a function is even function or odd function, first $ f\left( { - x} \right) $ by substituting $ - x $ for all the occurrence of $ x $ in $ f\left( x \right) $ .Now check If $ f\left( { - x} \right) = f\left( x \right) $ , then function is even and if $ f\left( { - x} \right) = - f\left( x \right) $ then the function is odd. And if neither the condition is true then the function is neither even nor odd.

Complete step-by-step answer:
 We are given a function in variable $ x $ as
 $ f\left( x \right) = 3{x^5} - 4x + 3 $
To determine whether a function is an even function or a odd function , we have to find the value of $ f\left( { - x} \right) $ and
If $ f\left( { - x} \right) = f\left( x \right) $ , then the function is an even function and
If $ f\left( { - x} \right) = - f\left( x \right) $ , then the function is a odd function
So let’s find out $ f\left( { - x} \right) $ by substituting $ - x $ for all the occurrence of $ x $ in $ f\left( x \right) $
 $
  f\left( { - x} \right) = 3{\left( { - x} \right)^5} - 4\left( { - x} \right) + 3 \\
   = 3{\left( { - 1} \right)^5}{\left( x \right)^5} - 4\left( { - 1} \right)\left( x \right) + 3 \\
  $
As we know negative number raised to power odd exponent is also negative in nature, so $ {\left( { - 1} \right)^5} = - 1\,and\,\left( { - 1} \right) = - 1 $
 $ f\left( { - x} \right) = - 3{x^5} + 4x + 3 $
As we can clearly see that neither condition is satisfied.
Therefore the given function is neither even nor odd.

Note: A function is basically a relation that results into a different output for each and every different input.
We can alternatively determine if a function is even or odd as If all the terms of a polynomial function are of odd degree then the function is odd and similarly if all the terms are even then the function is even.