Determine if the following function is even, odd, or neither. f(x) = -9x⁴+ 5x + 3

Solution:

Generally, we consider a real-valued function to be even or odd.

To identify if a function is even or odd, we plug in -x in place of x into the function f(x), that is, we check the output value of f(-x) to determine the type of the function.

Even and odd functions are symmetrical.

A function is said to be even if f(-x) = f(x)

A function is said to be odd if f(-x) = - f(x)

Given, f(x) = -9x⁴+ 5x + 3

f(-x) = -9(-x)⁴+ 5(-x) + 3

f(-x) = -9x⁴- 5x + 3

f(-x) ≠ f(x)

So, f(x) = -9x⁴+ 5x + 3 is not even.

f(-x) = -9(-x)⁴+ 5(-x) + 3

f(-x) = -9x⁴- 5x + 3

f(-x) ≠ - f(x)

So, f(x) = -9x⁴+ 5x + 3 is not odd.

Therefore, f(x) = -9x⁴+ 5x + 3 is neither even nor odd.