Question

Which of the following is the end behavior? is the degree of the function even, odd or neither?

Answer 1

The end behavior of a function describes how the function behaves as x approaches infinity or negative infinity. Even functions are symmetrical about the y-axis and odd functions have rotational symmetry about the origin. Examples include x²e-x² for even functions and xe-x² for odd functions.

Step-by-step explanation:

Understanding the End Behavior and Degree of Functions

The end behavior of a function refers to the behavior of the function values (y-values) as the input (x-values) either increase without bound (x → ∞) or decrease without bound (x → -∞). To determine whether the degree of a function is even or odd, we first need to understand the basic characteristics of even and odd functions. An even function is symmetrical about the y-axis and satisfies the condition y(x) = y(-x). In contrast, an odd function possesses rotational symmetry about the origin and satisfies the condition y(x) = −y(-x).

A simple example of an even function is the product x² e-x² (even times even is even), whereas xe-x² is an example of an odd function because it is the product of an odd function (x) and an even function (e-x²). The integral of an odd function over all space is zero, due to the symmetry of the function's values above and below the x-axis. Remember that determining the end behavior also depends on the leading coefficient and the degree of the polynomial; for instance, if the function has an even degree and a positive leading coefficient, the end behavior is such that as x approaches infinity or negative infinity, the function value will also approach infinity.

Answer 2

Degree

- We have that a function is odd if, for each x in the domain of f, f (- x) = - f( x ). Odd functions have rotational symmetry of 180º with respect to the origin.

- A function is even if, for each x in the domain of f, f (- x ) = f ( x ). Even functions have reflective symmetry across the y-axis.

Therefore, the degree of the function is neither.

End behavior

The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. So:

`\begin{gathered} f(x)\rightarrow\infty\text{, as x}\rightarrow\infty \\ \text{and} \\ f(x)\rightarrow-\infty,\text{ as x}\rightarrow-\infty \end{gathered}`

Answer:

9. Neither

10.

`\begin{gathered} as\text{ x}\rightarrow-\infty,f(x)\rightarrow-\infty \\ \text{as x}\rightarrow\infty,f(x)\rightarrow\infty \end{gathered}`