If R ( X ) = 3 X − 1 R(x)=3x-1 R ( X ) = 3 X − 1 And S ( X ) = 2 X + 1 S(x)=2x+1 S ( X ) = 2 X + 1 , Which Expression Is Equivalent To ( R S ) ( 6 \left(\frac{r}{s}\right)(6 ( S R ) ( 6 ]?A. 3 ( 6 ) − 1 2 ( 6 ) + 1 \frac{3(6)-1}{2(6)+1} 2 ( 6 ) + 1 3 ( 6 ) − 1 B. ( 6 ) 2 ( 6 ) + 1 \frac{(6)}{2(6)+1} 2 ( 6 ) + 1 ( 6 ) C. 36 − 1 26 + 1 \frac{36-1}{26+1} 26 + 1 36 − 1 D.

Mar 11, 2025 by ADMIN 379 views

If $r(x)=3x-1$ and $s(x)=2x+1$ , which expression is equivalent to $\left(\frac{r}{s}\right)(6)$ ?

Understanding the Problem

To find the equivalent expression for $\left(\frac{r}{s}\right)(6)$ , we need to understand the given functions $r(x)$ and $s(x)$ . The function $r(x)$ is defined as $3x-1$ , and the function $s(x)$ is defined as $2x+1$ . We are asked to find the expression equivalent to the ratio of these two functions evaluated at $x=6$ .

Evaluating the Functions

To evaluate the functions at $x=6$ , we substitute $x=6$ into the expressions for $r(x)$ and $s(x)$ .

$r(6) = 3(6) - 1 = 18 - 1 = 17$

$s(6) = 2(6) + 1 = 12 + 1 = 13$

Finding the Ratio

Now that we have evaluated the functions at $x=6$ , we can find the ratio of $r(6)$ to $s(6)$ .

$\left(\frac{r}{s}\right)(6) = \frac{r(6)}{s(6)} = \frac{17}{13}$

Evaluating the Options

We are given four options to choose from, and we need to determine which one is equivalent to the ratio we found.

A. $\frac{3(6)-1}{2(6)+1} = \frac{18-1}{12+1} = \frac{17}{13}$

B. $\frac{(6)}{2(6)+1} = \frac{6}{12+1} = \frac{6}{13}$

C. $\frac{36-1}{26+1} = \frac{35}{27}$

D. This option is not provided.

Conclusion

Based on our evaluation, we can see that option A is equivalent to the ratio we found. Therefore, the correct answer is A.

Discussion

This problem requires a basic understanding of algebra and function evaluation. The key concept is to substitute the given value of $x$ into the expressions for $r(x)$ and $s(x)$ and then find the ratio of the resulting values. This problem can be used to assess a student's ability to evaluate functions and find ratios.

Key Concepts

Function evaluation
Ratio of functions
Algebra

Related Topics

Evaluating functions at specific values
Finding ratios of functions
Algebraic expressions

Practice Problems

Evaluate the function $f(x) = 2x^2 + 3x - 1$ at $x = 4$ .
Find the ratio of the functions $g(x) = x^2 + 2x - 1$ and $h(x) = 2x^2 - 3x + 1$ at $x = 3$ .

Solutions

$f(4) = 2(4)^2 + 3(4) - 1 = 32 + 12 - 1 = 43$
$\frac{g(3)}{h(3)} = \frac{(3)^2 + 2(3) - 1}{2(3)^2 - 3(3) + 1} = \frac{9 + 6 - 1}{18 - 9 + 1} = \frac{14}{10} = \frac{7}{5}$
Q&A: Evaluating Functions and Finding Ratios

Q: What is the difference between evaluating a function and finding a ratio of functions?

A: Evaluating a function involves substituting a specific value into the expression for the function, whereas finding a ratio of functions involves dividing one function by another at a specific value.

Q: How do I evaluate a function at a specific value?

A: To evaluate a function at a specific value, substitute the value into the expression for the function and simplify the resulting expression.

Q: What is the key concept in finding a ratio of functions?

A: The key concept in finding a ratio of functions is to divide one function by another at a specific value.

Q: How do I find the ratio of two functions?

A: To find the ratio of two functions, divide the expression for one function by the expression for the other function at a specific value.

Q: What is the difference between a function and a ratio of functions?

A: A function is an expression that takes an input value and produces an output value, whereas a ratio of functions is the result of dividing one function by another at a specific value.

Q: Can I use the same method to evaluate a function and find a ratio of functions?

A: Yes, you can use the same method to evaluate a function and find a ratio of functions. The only difference is that you need to divide one function by another when finding a ratio.

Q: How do I choose the correct option when evaluating a function or finding a ratio of functions?

A: To choose the correct option, make sure to evaluate the function or find the ratio of functions at the specific value given in the problem.

Q: What are some common mistakes to avoid when evaluating functions and finding ratios of functions?

A: Some common mistakes to avoid include:

Not substituting the correct value into the expression for the function
Not simplifying the resulting expression
Not dividing one function by another when finding a ratio
Not choosing the correct option

Q: How can I practice evaluating functions and finding ratios of functions?

A: You can practice evaluating functions and finding ratios of functions by working through example problems and exercises. You can also use online resources and practice tests to help you prepare.

Q: What are some real-world applications of evaluating functions and finding ratios of functions?

A: Evaluating functions and finding ratios of functions have many real-world applications, including:

Modeling population growth and decline
Analyzing financial data
Predicting weather patterns
Optimizing business processes

Q: Can I use technology to help me evaluate functions and find ratios of functions?

A: Yes, you can use technology to help you evaluate functions and find ratios of functions. Many graphing calculators and computer algebra systems can help you evaluate functions and find ratios of functions.

Q: How can I use evaluating functions and finding ratios of functions in my career?

A: Evaluating functions and finding ratios of functions are essential skills in many careers, including:

Data analysis
Business management
Engineering
Science
Finance

Q: What are some common mistakes to avoid when using technology to evaluate functions and find ratios of functions?

A: Some common mistakes to avoid when using technology to evaluate functions and find ratios of functions include:

Not using the correct software or calculator
Not entering the correct values or expressions
Not understanding the output or results
Not using the technology correctly to evaluate functions and find ratios of functions

Q: How can I stay up-to-date with the latest developments in evaluating functions and finding ratios of functions?

A: You can stay up-to-date with the latest developments in evaluating functions and finding ratios of functions by:

Reading academic journals and research papers
Attending conferences and workshops
Joining online communities and forums
Following experts and thought leaders in the field

Q: Can I use evaluating functions and finding ratios of functions to solve real-world problems?

A: Yes, you can use evaluating functions and finding ratios of functions to solve real-world problems. Many real-world problems involve evaluating functions and finding ratios of functions, and using these skills can help you develop creative solutions to complex problems.