1.) Compute the domain of the function $$f(x)=\frac{1}{\lfloor x^2-7x+13\rfloor}.$$

2.) Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \[f(n) = \left\{\begin{array}{cl} 6 & \text{ if }n\text{ is a multiple of 2 and 3}, \\2 & \text{ if }n\text{ is only a multiple of 2}, \\0 & \text{ if }n\text{ is not a multiple of 2}.\end{array}\right.\]Allie rolls the die four times and gets a 5, 4, 1, and 2. Betty rolls and gets 6, 3, 3, and 2. What is the product of Allie's total points and Betty's total points?

lolzforlife Apr 27, 2019

#1**+4 **

\(f(x)=\dfrac{1}{\lfloor x^2-7x+13\rfloor}\)

\(\text{The floor function will map }[0,1) \to 0\\ \text{which results in undefined division by 0}\\ \text{Thus the interval }[0,1) \text{ must be restricted from the floor function}\\ \text{thus }x^2-7x+13<0 \text{ OR } x^2-7x+13 \geq 1\)

\(x^2 - 7x+13 < 0\\ \left(x-\dfrac 7 2\right)^2 +\dfrac 3 4 < 0\\ \text{this clearly cannot occur with real numbers}\)

\(x^2 - 7x+13 \geq 1\\ x^2 - 7x + 12 \geq 0\\ \left(x-\dfrac 7 2\right)^2 - \dfrac 1 4 \geq 0\\ \left(x-\dfrac 7 2\right)^2 \geq \dfrac 1 4\\ \left(x-\dfrac 7 2\right) \geq \dfrac 1 2 \text{ OR }\left(x-\dfrac 7 2\right) \leq -\dfrac 1 2\\ x \geq 4 \text{ OR }x \leq 3\\ x \in (-\infty, 3]\cup [4,\infty)\)

.Rom Apr 27, 2019

#3**+2 **

How about thanking Rom for the answer you already have.

Or asking him questions if your do not fully understand?

Melody
Apr 28, 2019

#4**+1 **

Sorry for sounding impatient. I figured out number 2 so it no longer has to be answered! Thanks @Rom and @Melody.

lolzforlife
Apr 28, 2019