Question

b) for the two sets below, please map a function between them. set x: algebra, geometry, statistics, calculus, arithmetic. set y: red, blue, yellow, green, orange. if a polynomial function is even, what can be said about the highest - the function? bonus: what can be said about the function as it goes to + and -? if f(x)=x^4 + x^2 + x - 1, what is f(-2)?

Explanation:

Step1: Define the function mapping

Let \(f(x)\) be the mapping function. We can define \(f\) such that \(f(\text{Algebra})=\text{Red}\), \(f(\text{Geometry})=\text{Blue}\), \(f(\text{Statistics})=\text{Yellow}\), \(f(\text{Calculus})=\text{Green}\), \(f(\text{Arithmetic})=\text{Orange}\).

Step2: Analyze the polynomial - related questions

For the question about even - polynomial functions

An even polynomial function \(y = f(x)\) satisfies \(f(x)=f( - x)\) for all \(x\) in the domain. The highest - degree term of an even polynomial function has an even exponent. For example, if \(f(x)=a_nx^n+\cdots+a_1x + a_0\) is even, then \(n\) is even.

For the bonus question

As \(x\to+\infty\) and \(x\to-\infty\) for an even polynomial function \(y = f(x)=a_nx^n+\cdots+a_0\) (\(n\) even, \(a_n
eq0\)), if \(a_n>0\), then \(y\to+\infty\) as \(x\to\pm\infty\); if \(a_n < 0\), then \(y\to-\infty\) as \(x\to\pm\infty\).

For the function - evaluation question

Given \(f(x)=x^4 + x^2+x - 1\), substitute \(x=-2\) into the function.
\[

$$\begin{align*} f(-2)&=(-2)^4+(-2)^2+(-2)-1\\ &=16 + 4-2-1\\ &=17 \end{align*}$$

\]

Answer:

The function mapping: \(f(\text{Algebra})=\text{Red}\), \(f(\text{Geometry})=\text{Blue}\), \(f(\text{Statistics})=\text{Yellow}\), \(f(\text{Calculus})=\text{Green}\), \(f(\text{Arithmetic})=\text{Orange}\); For an even polynomial function, the highest - degree term has an even exponent; For the bonus: if the leading coefficient of an even polynomial is positive, the function approaches \(+\infty\) as \(x\to\pm\infty\), if negative, it approaches \(-\infty\) as \(x\to\pm\infty\); \(f(-2)=17\)