Solving (r/s)(6) With R(x)=3x-1, S(x)=2x+1

Hey math whizzes! Today, we're diving into a super cool problem involving function notation and operations. We're given two functions, $r(x) = 3x - 1$ and $s(x) = 2x + 1$, and we need to figure out which expression is equivalent to $\left(\frac{r}{s}\right)(6)$. This might sound a bit intimidating at first, but trust me, guys, once we break it down, it's totally manageable. We're going to explore how to combine functions and evaluate them at a specific point. So, grab your pencils, get comfortable, and let's unravel this mathematical puzzle together. We'll go step-by-step, making sure we understand each part of the process. By the end of this, you'll be a pro at function division and evaluation!

Understanding Function Operations: The Division Part

Alright, let's kick things off by understanding what $\left(\frac{r}{s}\right)(x)$ actually means. When we see this notation, it tells us we're performing a division operation between two functions, $r(x)$ and $s(x)$. Specifically, it means we take the function $r(x)$ and divide it by the function $s(x)$. So, in terms of our given functions, we have $r(x) = 3x - 1$ and $s(x) = 2x + 1$. Therefore, $\left(\frac{r}{s}\right)(x)$ can be written as $\frac{r(x)}{s(x)}$. Substituting the expressions for $r(x)$ and $s(x)$, we get $\frac{3x - 1}{2x + 1}$. This is the general form of the combined function through division. Remember, for this expression to be defined, the denominator, $s(x)$, cannot be equal to zero. So, $2x + 1 \neq 0$, which means $x \neq -1/2$. This is an important detail to keep in mind when working with rational functions. Now that we've got the general form down, the next step is to evaluate this new function at a specific value of $x$.

Evaluating the Combined Function at x = 6

Now that we've established that $\left(\frac{r}{s}\right)(x) = \frac{3x - 1}{2x + 1}$, the problem asks us to find the value of this expression when $x = 6$. This is where we substitute the value 6 for every instance of $x$ in our combined function. So, we'll replace $x$ with 6 in the numerator and the denominator. Plugging in $x = 6$ into the numerator, $3x - 1$, we get $3(6) - 1$. And plugging in $x = 6$ into the denominator, $2x + 1$, we get $2(6) + 1$. So, the expression $\left(\frac{r}{s}\right)(6)$ becomes $\frac{3(6) - 1}{2(6) + 1}$. This is the exact value we're looking for. The problem, however, isn't asking for the final numerical answer, but rather which of the given options is equivalent to this expression. So, our task is to compare this form $\frac{3(6) - 1}{2(6) + 1}$ with the provided choices (A, B, C, and D) and see which one matches.

Comparing Our Result with the Options

Let's take a good look at the options provided and see which one perfectly aligns with our derived expression $\frac{3(6) - 1}{2(6) + 1}$.

• Option A: $\frac{3(6)-1}{2(6)+1}$ - Wow, guys, this looks exactly like what we derived! It directly substitutes $x=6$ into the expressions for $r(x)$ and $s(x)$ before performing any calculations. This is a strong contender.

• Option B: $\frac{(6)}{2(6)+1}$ - This option seems to have replaced $r(x)$ with just $(6)$. This isn't correct because $r(x)$ is $3x-1$, not just $x$. So, this option is out.

• Option C: $\frac{36-1}{26+1}$ - This option looks like it might have done some calculation, but it seems to have made errors. For example, $3(6)$ is $18$, not $36$. Also, $2(6)$ is $12$, not $26$. So, this option is definitely incorrect.

• Option D: $\frac{(6)-1}{(6)+1}$ - This option seems to have replaced $3x-1$ with $(6)-1$ and $2x+1$ with $(6)+1$. This is also not correct. It looks like it might have simplified $r(x)$ and $s(x)$ incorrectly or applied the substitution in a strange way. This option is incorrect.

Based on our comparison, Option A is the only one that perfectly matches the expression we obtained by substituting $x=6$ into $\frac{r(x)}{s(x)}$. It represents the direct substitution of the value into the function definitions before any arithmetic simplification occurs, which is precisely what the question asks for: an equivalent expression.

The Final Calculation (Just for Fun!)

While the question only asks for the equivalent expression, let's go ahead and calculate the actual value of $\left(\frac{r}{s}\right)(6)$ to satisfy our curiosity. We have the expression from Option A: $\frac{3(6)-1}{2(6)+1}$.

First, let's evaluate the numerator: $3(6) - 1 = 18 - 1 = 17$.

Next, let's evaluate the denominator: $2(6) + 1 = 12 + 1 = 13$.

So, the numerical value of $\left(\frac{r}{s}\right)(6)$ is $\frac{17}{13}$. This fraction cannot be simplified further. So, if the question had asked for the value, the answer would be $\frac{17}{13}$. But since it asked for the equivalent expression, Option A is our clear winner.

Key Takeaways and Practice

This problem highlights a few crucial concepts in algebra, guys. Firstly, it reinforces our understanding of function notation. Remember that $r(x)$ and $s(x)$ are just names for expressions that depend on the variable $x$. Secondly, it demonstrates function operations, specifically division. The notation $\left(\frac{r}{s}\right)(x)$ is a compact way to represent $\frac{r(x)}{s(x)}$. Finally, it emphasizes the importance of evaluating functions at specific points. When asked to evaluate $\left(\frac{r}{s}\right)(6)$, we need to substitute 6 for $x$ in both $r(x)$ and $s(x)$ before performing any arithmetic. The key is to look for an expression that mirrors this substitution process. If you encounter similar problems, always break down the notation, identify the operations involved, and then perform the substitution systematically. Don't get tricked by options that perform calculations prematurely or substitute incorrectly. Practice makes perfect, so try to find more problems like this and work through them. You'll get faster and more confident with each one you solve! Keep up the great work, and happy calculating!