Question

1. 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. 7) if a polynomial function is even, what can be said about the highest exponent the function? bonus: what can be said about the function as it goes to + and - infinity? if f(x)=x^4 + x^2 + x - 1, what is f(-2)? what is a relation? mapping between 2 sets what is a function? where each input has exactly one output what is the domain? set your mapping from (input) what is the range? set your mapping to (output) what is the vertical line test?

Explanation:

6)

A function can be defined as \(f(x)\) where \(x\in Set\ X\) and \(f(x)\in Set\ Y\). For example, \(f(Algebra) = Red\), \(f(Geometry)=Blue\), \(f(Statistics) = Yellow\), \(f(Calculus)=Green\), \(f(Arithmetic)=Orange\).

7)

• Main part:
• An even - polynomial function \(f(x)\) satisfies \(f(x)=f( - x)\). For a polynomial \(f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0\), if \(f(x)=f( - x)\), then all the terms with odd - powered exponents must have a coefficient of 0. So, the highest exponent \(n\) of an even polynomial function is an even number.
• Bonus:
• Let \(y = f(x)\) be an even polynomial function \(y=a_nx^n+\cdots+a_0\) with \(n\) even and \(a_n

eq0\).

• As \(x\to+\infty\), if \(a_n>0\), then \(y\to+\infty\); if \(a_n < 0\), then \(y\to-\infty\).
• As \(x\to-\infty\), since \(y = f(x)=f( - x)\) for an even function, if \(a_n>0\), then \(y\to+\infty\); if \(a_n < 0\), then \(y\to-\infty\). In other words, for an even - polynomial function \(y = f(x)\), \(\lim_{x\to+\infty}f(x)=\lim_{x\to-\infty}f(x)\).

8)

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

• First, calculate \((-2)^4=16\), \((-2)^2 = 4\).
• Then \(f(-2)=(-2)^4+(-2)^2+(-2)-1=16 + 4-2 - 1=17\).

9)

A relation is a mapping between two sets.

10)

A function is a relation where each input has exactly one output.

11)

The domain is the set of all inputs (the set from which the mapping starts).

12)

The range is the set of all outputs (the set to which the mapping goes).

13)

The vertical - line test is a graphical method used to determine if a curve in the \(xy\) - plane represents a function. If any vertical line intersects the curve at most once, then the curve represents a function.

Answer:

• 6) \(f(Algebra) = Red\), \(f(Geometry)=Blue\), \(f(Statistics) = Yellow\), \(f(Calculus)=Green\), \(f(Arithmetic)=Orange\)
• 7) The highest exponent of an even polynomial function is an even number. As \(x\to+\infty\) and \(x\to-\infty\), for an even - polynomial function \(y = f(x)\), \(\lim_{x\to+\infty}f(x)=\lim_{x\to-\infty}f(x)\) (if the leading coefficient \(a_n>0\), \(y\to+\infty\) as \(x\to\pm\infty\); if \(a_n < 0\), \(y\to-\infty\) as \(x\to\pm\infty\))
• 8) 17
• 9) A mapping between two sets
• 10) A relation where each input has exactly one output
• 11) The set of all inputs
• 12) The set of all outputs
• 13) A graphical method to determine if a curve represents a function. If any vertical line intersects the curve at most once, then the curve represents a function.