Question

State whether the following statement is true or false. If the statement is false, re-write it to make it true.If the polynomial has an even degree, then it is an even function.

Answer 1

False.

If the polynomial has an even degree, then it is not necessarily an even function.

Explanation

Let's take for example the next even degree polynomial:

`f(x)=x^2`

In even functions, the next condition must be satisfied:

`f(x)=f(-x)`

Evaluating f(-x) in this case:

`\begin{gathered} f(-x)=(-x)^2 \\ f(-x)=x^2\text{ \lparen the square of a negative number is equal to the square of its opposite\rparen} \end{gathered}`

Then, f(x) = x² is even.

Now, let's analyze the next even degree polynomial:

`f(x)=(x-2)^2`

Evaluating this function at x = -1 and x = 1, we get:

`\begin{gathered} f(-1)=(-1-2)^2 \\ f(-1)=(-3)^2 \\ f(-1)=9 \\ f(1)=(1-2)^2 \\ f(1)=(-1)^2 \\ f(1)=1 \\ \text{ Therefore:} \\ f(-1)f(1) \end{gathered}`

In conclusion, f(x) = (x - 2)² is not even.