Question

state if the following functions are even, odd, or neither.

1. $f(x)=4x^{7}+5x^{3}-2x$
2. $f(x)=7 - 6x^{4}-3x^{2}$

describe the end - behavior of each function using limit notation.

1. $p(x)=-11x^{7}-6x^{2}+4x$
2. $p(x)=-7x^{6}+4x - 8$

34.

1. sketch the graph of a polynomial function that could match statements $lim_{x

ightarrow-infty}p(x)=infty$ and $lim_{x
ightarrowinfty}p(x)=infty$.

Explanation:

Step1: Recall even - odd function definitions

An even function satisfies $f(-x)=f(x)$ and an odd function satisfies $f(-x)=-f(x)$.

Step2: Check function 30 ($f(x) = 4x^{7}+5x^{3}-2x$)

Calculate $f(-x)$:
\[

$$\begin{align*} f(-x)&=4(-x)^{7}+5(-x)^{3}-2(-x)\\ &=- 4x^{7}-5x^{3}+2x\\ &=-(4x^{7}+5x^{3}-2x)\\ &=-f(x) \end{align*}$$

\]
So, $f(x)$ is odd.

Step3: Check function 31 ($f(x)=7 - 6x^{4}-3x^{2}$)

Calculate $f(-x)$:
\[

$$\begin{align*} f(-x)&=7-6(-x)^{4}-3(-x)^{2}\\ &=7 - 6x^{4}-3x^{2}\\ &=f(x) \end{align*}$$

\]
So, $f(x)$ is even.

Step4: Analyze end - behavior of function 32 ($p(x)=-11x^{7}-6x^{2}+4x$)

The leading term is $-11x^{7}$. As $x\to\infty$, $p(x)=-11x^{7}(1 +\frac{6}{11x^{5}}-\frac{4}{11x^{6}})\to-\infty$ and as $x\to-\infty$, $p(x)=-11x^{7}(1 +\frac{6}{11x^{5}}-\frac{4}{11x^{6}})\to\infty$. In limit notation: $\lim_{x\to\infty}p(x)=-\infty$ and $\lim_{x\to-\infty}p(x)=\infty$.

Step5: Analyze end - behavior of function 33 ($p(x)=-7x^{6}+4x - 8$)

The leading term is $-7x^{6}$. As $x\to\pm\infty$, $p(x)=-7x^{6}(1-\frac{4}{7x^{5}}+\frac{8}{7x^{6}})\to-\infty$. In limit notation: $\lim_{x\to\infty}p(x)=-\infty$ and $\lim_{x\to-\infty}p(x)=-\infty$.

Step6: Sketch function for 35

A polynomial function with $\lim_{x\to-\infty}p(x)=\infty$ and $\lim_{x\to\infty}p(x)=\infty$ has an even - degree and a positive leading coefficient. For example, a simple quadratic function $y = x^{2}$ has this property. A more general polynomial like $y=(x - 1)(x + 1)(x-2)(x + 2)+10$ (a fourth - degree polynomial) would also work. The graph is a smooth curve that opens upwards, with no specific roots or turning points required to be found precisely for a sketch. It should be symmetric about a vertical line (for an even - degree polynomial) and go to positive infinity as $x$ moves in both the positive and negative directions.

Answer:

1. The function $f(x) = 4x^{7}+5x^{3}-2x$ is odd.
2. The function $f(x)=7 - 6x^{4}-3x^{2}$ is even.
3. $\lim_{x\to\infty}p(x)=-\infty$, $\lim_{x\to-\infty}p(x)=\infty$
4. $\lim_{x\to\infty}p(x)=-\infty$, $\lim_{x\to-\infty}p(x)=-\infty$
5. Sketch a smooth, upward - opening curve (even - degree polynomial with positive leading coefficient) that goes to $\infty$ as $x\to\pm\infty$.