Problem 101 - Algebra for College Students

Question

Use a calculator to find \(R(2), R(30), R(500)\) \(R(9000),\) and \(R(80,000)\) for the rational function $$ R(x)=\frac{x-3}{2 x+1} $$ Round answers to four decimal places. What can you conclude about the value of \(R(x)\) as \(x\) gets larger and larger without bound?

Step-by-Step Solution

Answer

R(2) = -0.2000, R(30) = 0.4426, R(500) = 0.4965, R(9000) = 0.4998, R(80,000) = 0.49998. As x increases, R(x) approaches 0.5.

Step 1: Find R(2)

Substitute 2 into the function. Compute using a calculator as follows: \[ R(2) = \frac{2-3}{2(2)+1} = \frac{-1}{5} = -0.2000 \].

Step 2: Find R(30)

Substitute 30 into the function. Compute using a calculator as follows: \[ R(30) = \frac{30-3}{2(30)+1} = \frac{27}{61} \ = 0.4426 \].

Step 3: Find R(500)

Substitute 500 into the function. Compute using a calculator as follows: \[ R(500) = \frac{500-3}{2(500)+1} = \frac{497}{1001} \ = 0.4965 \].

Step 4: Find R(9000)

Substitute 9000 into the function. Compute using a calculator as follows: \[ R(9000) = \frac{9000-3}{2(9000)+1} = \frac{8997}{18001} \ = 0.4998 \].

Step 5: Find R(80,000)

Substitute 80,000 into the function. Compute using a calculator as follows: \[ R(80000) = \frac{80000-3}{2(80000)+1} = \frac{79997}{160001} \ = 0.49998 \].

Step 6: Conclusion

As \( x \) gets larger and larger without bound, the value of \( R(x) \) approaches 0.5. This is because, for very large values of \( x \), the -3 and +1 become insignificant, making the function \[ R(x) \ \frac{x}{2x} \ =0.5 \].

Key Concepts

Evaluating Functions

When working with functions, evaluating them at specific points helps us understand their behavior. In this exercise, we evaluate the rational function \( R(x) = \frac{x-3}{2x+1} \) at different values of x. To do this, we simply substitute the given x-values into the function and use a calculator to perform the arithmetic. For instance, substituting 2 into the function gives: \[ R(2) = \frac{2-3}{2(2)+1} = \frac{-1}{5} = -0.2000 \]. By doing this for several values, we observe how the output of the function changes.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as the input grows very large or very small. In this exercise, as x gets larger, we see that \( R(x)\) approaches 0.5. This is because, for large values of x, the -3 and +1 in the numerator and denominator become negligible. Thus, the function simplifies to: \[ R(x) \approx \frac{x}{2x} = 0.5 \]. This shows that the function has a horizontal asymptote at y = 0.5. Understanding asymptotic behavior is crucial, as it provides insights into the function's long-term trends.

Calculators in Mathematics

Calculators are indispensable tools in mathematics. They help us quickly and accurately compute complex expressions, especially when working with irrational functions. In this exercise, using a calculator ensures we get precise values like \( R(30) = 0.4426 \) and \( R(500) = 0.4965 \). To effectively use a calculator: \ * Begin by entering the expression correctly.
* Use parentheses to ensure proper order of operations.
* Round your answers to the required decimal places.
Practicing with calculators can significantly improve your problem-solving speed and accuracy.

Algebra

Algebra forms the foundation for understanding and solving equations involving variables. In this exercise, we use algebraic manipulation to simplify and evaluate the function \( R(x) = \frac{x-3}{2x+1} \). Key concepts include:
* Substitution: Replacing variables with given values to find specific outputs.
* Simplification: Reducing expressions to simplest forms.
* Interpreting results: Understanding the meaning behind computed values.
Mastering these algebraic techniques is crucial for tackling more advanced mathematical concepts and functions.