Math Problem: Simplify Rational Function (r/s)(b)

Hey guys, let's dive into a common algebra problem that pops up in math class! We've got two functions, $r(x) = 3x - 1$ and $s(x) = 2x + 1$. The question asks us to find an expression equivalent to $\left(\frac{r}{s}\right)(b)$. This notation might look a little intimidating at first, but trust me, it's all about understanding function notation and how to combine functions. We'll break it down step-by-step so you can tackle similar problems with confidence. Get ready to flex those math muscles!

Understanding Function Notation and Operations

Before we even look at the options, let's get a solid grasp on what we're dealing with. The notation $r(x)$ and $s(x)$ simply means that 'r' and 's' are functions of the variable 'x'. So, $r(x) = 3x - 1$ means that for any input 'x', the function 'r' multiplies it by 3 and then subtracts 1. Similarly, $s(x) = 2x + 1$ means that for any input 'x', the function 's' multiplies it by 2 and then adds 1. Now, what about $\left(\frac{r}{s}\right)(b)$? This is asking us to consider the division of the function $r$ by the function $s$, and then evaluate this new combined function at a specific value, which is 'b' in this case. Remember, when we divide two functions, say $f(x)$ and $g(x)$, we write it as $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, with the condition that $g(x) \neq 0$. So, for our problem, $\left(\frac{r}{s}\right)(x) = \frac{r(x)}{s(x)}$. Since we need to evaluate this at 'b', we simply replace every 'x' with 'b'. This gives us $\left(\frac{r}{s}\right)(b) = \frac{r(b)}{s(b)}$. Now, we just need to substitute the expressions for $r(x)$ and $s(x)$ with 'b' as the input. For $r(b)$, we take the rule for $r(x)$ and plug in 'b': $r(b) = 3(b) - 1$. For $s(b)$, we do the same: $s(b) = 2(b) + 1$. Putting it all together, we get $\left(\frac{r}{s}\right)(b) = \frac{3b - 1}{2b + 1}$. This is our target expression. Keep this in mind as we go through the multiple-choice options provided!

Analyzing the Multiple-Choice Options

Alright guys, we've figured out the core of the problem: $\left(\frac{r}{s}\right)(b) = \frac{3b - 1}{2b + 1}$. Now, let's scrutinize each of the given options to see which one matches our result. Remember, we're looking for an expression that is equivalent to $\frac{3b - 1}{2b + 1}$.

• Option A: $\frac{3(6)-1}{2(6)+1}$ Looking at this option, we see the number '6' plugged in for 'b'. This would be equivalent to $\left(\frac{r}{s}\right)(6)$, not $\left(\frac{r}{s}\right)(b)$. While the form of the expression (a numerator and a denominator) is correct, the specific value used (6 instead of b) makes it incorrect for the general expression we need. It's a common distractor, trying to get you to plug in a number prematurely.

• Option B: $\frac{(6)}{2(6)+1}$ This option seems to be a mix-up. The denominator $2(6)+1$ correctly represents $s(6)$, but the numerator, which is simply '(6)', doesn't match $r(6) = 3(6) - 1$. So, this option is definitely not equivalent to our derived expression.

• Option C: $\frac{36-1}{26+1}$ This one is a bit trickier and relies on a common mistake. It looks like someone might have tried to substitute 'b' with '6' and then perhaps misinterpreted the multiplication. For instance, $36-1$ is not $3(6)-1$, and $26+1$ is not $2(6)+1$. If we were to interpret $36$ as $3 imes 6$ and $26$ as $2 imes 6$, then this option might seem plausible at first glance. However, the way it's written, $36$ is just the number thirty-six, not three times six. This is a clear indicator of a misinterpretation or a typo, making it incorrect.

• Option D: $\frac{(6)-1}{(6)+1}$ Similar to Option A, this option uses the number '6' instead of the variable 'b'. It appears to be attempting to represent $\frac{r(6)}{s(6)}$ but gets the numerator wrong. $r(6)$ should be $3(6)-1$, not $(6)-1$. This option is also incorrect because it uses a specific value and a misrepresentation of the function $r(x)$.

Wait a minute! It seems there might be a misunderstanding or a typo in the provided options based on our derived correct expression $\frac{3b - 1}{2b + 1}$. Let's re-examine the problem and options very carefully. The original problem states $r(x)=3x-1$ and $s(x)=2x+1$, and asks for $\left(\frac{r}{s}\right)(b)$. We correctly deduced this should be $\frac{r(b)}{s(b)} = \frac{3b-1}{2b+1}$.

Let's assume there was a typo in the question itself, and it was intended to ask for $\left(\frac{r}{s}\right)(6)$, not $\left(\frac{r}{s}\right)(b)$. If that were the case, then:

$r(6) = 3(6) - 1 = 18 - 1 = 17$ $s(6) = 2(6) + 1 = 12 + 1 = 13$

So, $\left(\frac{r}{s}\right)(6) = \frac{17}{13}$. Now let's look at the options again:

• Option A: $\frac{3(6)-1}{2(6)+1}$ This exactly matches our calculation for $\left(\frac{r}{s}\right)(6)$. The numerator is $3(6)-1$ and the denominator is $2(6)+1$. This is the direct substitution of $x=6$ into $\frac{r(x)}{s(x)}$.

• Option B: $\frac{(6)}{2(6)+1}$ Numerator is wrong ($6$ instead of $3(6)-1$).

• Option C: $\frac{36-1}{26+1}$ This represents $\frac{3 imes 10 + 6 - 1}{2 imes 10 + 6 + 1}$ or similar misinterpretations, not $\frac{17}{13}$.

• Option D: $\frac{(6)-1}{(6)+1}$ Numerator is wrong ($(6)-1$ instead of $3(6)-1$) and denominator is wrong ($(6)+1$ instead of $2(6)+1$).

Given this analysis, it's highly probable that the question intended to ask for the evaluation at $x=6$, or that the options provided are for a different but related question. However, strictly answering the question as written: "which expression is equivalent to $\left(\frac{r}{s}\right)(b)$?", the correct answer should be $\frac{3b-1}{2b+1}$. Since none of the options perfectly represent $\frac{3b-1}{2b+1}$ using the variable 'b', we must conclude that either there's a typo in the question (likely meaning to ask for $\left(\frac{r}{s}\right)(6)$) or a typo in the options. If we must choose from the given options and assume the question meant to ask for $\left(\frac{r}{s}\right)(6)$, then Option A is the correct representation of that calculation.

The Final Answer and Why

Let's be super clear here. The question as written asks for $\left(\frac{r}{s}\right)(b)$. We derived that this is equal to $\frac{3b-1}{2b+1}$. None of the options are written in terms of 'b'. However, Option A, $\frac{3(6)-1}{2(6)+1}$, perfectly represents the substitution for when the input is 6. This strongly suggests the question meant to ask for $\left(\frac{r}{s}\right)(6)$.

If we proceed under the assumption that the question implicitly intended to evaluate at $b=6$ (which is a common way these problems are posed, often with a specific numerical value for the variable), then Option A is the most direct and accurate representation of that calculation. It shows the numerator $r(6) = 3(6)-1$ and the denominator $s(6) = 2(6)+1$ explicitly.

Therefore, assuming the question implies evaluation at b=6, the equivalent expression is $\boxed{A. \frac{3(6)-1}{2(6)+1}}$. If the question strictly meant 'b' as a variable, then none of the options are correct as provided. Always double-check the problem statement and the options provided for any potential ambiguities or typos, guys!