# Thread: Critical Thinking with Functions

1. ## Critical Thinking with Functions

Hello, I'm new to the forum and would like to get some help with two problems that both relate to analyzing functions. I understand the basic concepts but these questions have been placed under the category of "critical thinking" and I'm not sure how to go about solving them.

Problem #1

If $\displaystyle f(2m+1)=24m^3+36m^2+26m$, what is $\displaystyle f(x)$?
(Hint: Begin by solving $\displaystyle x=2m+1$ for $\displaystyle m$.)

I tried this, and so far I got $\displaystyle m= \frac {-1}{2}$ for the "hint". I'm not sure if that's what I was supposed to try to figure out, and if it is, I'm not sure where to go from that point.

Problem #2

$\displaystyle P(x)$ is a function for which $\displaystyle P(1)=1$, $\displaystyle P(2)=2$, $\displaystyle P(3)=3$, and $\displaystyle P(x+1)= \frac {P(x-2)P(x-1)+1}{P(x)}$ for $\displaystyle x \geq 3$. Find the value of $\displaystyle P(6)$.

To be honest, I don't know where to start with this one. I don't know how to deal with all the information that the question is giving me.

2. ## Re: Critical Thinking with Functions

Originally Posted by sararose
Hello, I'm new to the forum and would like to get some help with two problems that both relate to analyzing functions. I understand the basic concepts but these questions have been placed under the category of "critical thinking" and I'm not sure how to go about solving them.

Problem #1

If $\displaystyle f(2m+1)=24m^3+36m^2+26m$, what is $\displaystyle f(x)$?
(Hint: Begin by solving $\displaystyle x=2m+1$ for $\displaystyle m$.)

I tried this, and so far I got $\displaystyle m= \frac {-1}{2}$ for the "hint". I'm not sure if that's what I was supposed to try to figure out, and if it is, I'm not sure where to go from that point.
What happened to the "x"? m would equal -1/2 if x were equal to 0. Instead, from 2m- 1= x, 2m= x- 1, m= (x- 1)/2. Now replace m by that:
$\displaystyle f(2m+1)=24m^3+36m^2+26m$ so
$\displaystyle f(x)= 24\frac{(x-1)^3}{2^3}+ 36\frac{(x-1)^2}{2^2}+ 26\frac{x-1}{2}$.

Problem #2

$\displaystyle P(x)$ is a function for which $\displaystyle P(1)=1$, $\displaystyle P(2)=2$, $\displaystyle P(3)=3$, and $\displaystyle P(x+1)= \frac {P(x-2)P(x-1)+1}{P(x)}$ for $\displaystyle x \geq 3$. Find the value of $\displaystyle P(6)$.

To be honest, I don't know where to start with this one. I don't know how to deal with all the information that the question is giving me.
If x= 3, x+1= 4, x-1= 2, and x-2= 1 so according to that formula, $\displaystyle P(4)= \frac{P(1)P(2)+ 1}{P(3)}$. Now that you know P(4), what is P(5)?

3. ## Re: Critical Thinking with Functions

Thank you very much! Following your explanation for Question #2, P(5) would be $\displaystyle \frac {P(2)P(3)+1}{P(4)}$, and by succession P(6) would be $\displaystyle \frac {P(3)P(4)+1}{P(5)}$, correct?

4. ## Re: Critical Thinking with Functions

Originally Posted by sararose
Thank you very much! Following your explanation for Question #2, P(5) would be $\displaystyle \frac {P(2)P(3)+1}{P(4)}$, and by succession P(6) would be $\displaystyle \frac {P(3)P(4)+1}{P(5)}$, correct?
Yes, now what are the numerical values?