Question

2-133. algebraically determine if each of the following functions is even, odd, or neither. verify your answers graphically. homework help
a. $f(x)=3x^{2}+5x - 1$
b. $g(x)=\frac{3}{x^{2}}+5x^{4}-1$

Explanation:

Step1: Recall the definitions

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

Step2: Analyze function $f(x)=3x^{2}+5x - 1$

Calculate $f(-x)$:
\[

$$\begin{align*} f(-x)&=3(-x)^{2}+5(-x)-1\\ &=3x^{2}-5x - 1 \end{align*}$$

\]
Since $f(-x)
eq f(x)$ (because $3x^{2}-5x - 1
eq3x^{2}+5x - 1$) and $f(-x)
eq - f(x)$ (where $-f(x)=-3x^{2}-5x + 1$), $f(x)$ is neither even nor odd.

Step3: Analyze function $g(x)=\frac{3}{x^{2}}+5x^{4}-1$

Calculate $g(-x)$:
\[

$$\begin{align*} g(-x)&=\frac{3}{(-x)^{2}}+5(-x)^{4}-1\\ &=\frac{3}{x^{2}}+5x^{4}-1 \end{align*}$$

\]
Since $g(-x)=g(x)$, $g(x)$ is an even function.

Answer:

a. Neither
b. Even