Question Video: Stating the Parity of a Polynomial Function

Is the function 𝑓(𝑥) = (4𝑥⁴ + 5)² + 2 even, odd or neither even nor odd?

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Video Transcript

Is the function 𝑓 of 𝑥 equal to four 𝑥 to the power of four plus five all squared plus two even, odd, or neither even nor odd?

We recall that any real-valued function is even if 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥. The function is odd, on the other hand, if 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. In order to work out whether this function is even or odd, we need to find an expression for 𝑓 of negative 𝑥. We do this by replacing each 𝑥 in the function 𝑓 of 𝑥 with negative 𝑥. This gives us four multiplied by negative 𝑥 to the power of four plus five all squared plus two.

We know that negative 𝑥 squared is equal to 𝑥 squared, as multiplying a negative by a negative gives a positive answer. Cubing negative 𝑥 gives us negative 𝑥 cubed, as we are multiplying 𝑥 squared by negative 𝑥. It therefore follows that negative 𝑥 to the power of four is positive and is equal to 𝑥 to the power of four.

Our expression for 𝑓 of negative 𝑥 can therefore be rewritten as four 𝑥 to the power of four plus five all squared plus two. This is the same as our initial expression 𝑓 of 𝑥. As 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥, we can conclude that the function is even.

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