3. Give a symbolic example of a quartic function that is even and $$ f(x) \rightarrow-\infty $$ as $$ x \rightarrow \pm \infty $$.

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UpStudy AI · Accepted Answer

1. Since the function is even (i.e., $$ f(x)=f(-x) $$), it must contain only even powers of $$ x $$.  
2. A general quartic function with only even terms can be written as  
   $$
   f(x)= ax^4+ bx^2+ c.
   $$
3. The condition $$\displaystyle f(x) \rightarrow -\infty$$ as $$\displaystyle x \rightarrow \pm \infty$$ requires the highest degree term to have a negative coefficient. In this case, $$ ax^4 $$ dominates for large $$|x|$$, so we must have  
   $$
   a < 0.
   $$
4. A simple choice is to let $$ a=-1 $$ and ignore the lower order terms (or set $$ b=c=0 $$), which gives  
   $$
   f(x)= -x^4.
   $$
5. This function satisfies both properties: it is even and  
   $$
   f(x) \rightarrow -\infty \quad \text{as} \quad x \rightarrow \pm \infty.
   $$

Final answer:  
$$
f(x)= -x^4.
$$
$$ f(x) = -x^4 $$ is an even quartic function that approaches negative infinity as $$ x $$ approaches both positive and negative infinity.

3. Give a symbolic example of a quartic function that is even and \( f(x) \rightarrow-\infty \) as \( x \rightarrow \pm \infty \).

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