Problem 102 - Algebra for College Students

Question

Use a calculator to find \(H(1000), H(100,000)\) \(H(1,000,000),\) and \(H(10,000,000)\) for the rational function $$ H(x)=\frac{7 x-50}{3 x+91} $$ Round answers to four decimal places. What can you conclude about the value of \(H(x)\) as \(x\) gets larger and larger without bound?

Step-by-Step Solution

Answer

As \(x\) gets larger, \(H(x)\rightarrow \frac{7}{3}\approx 2.3333\).

Step 1: Understand the Rational Function

The given function is \(H(x) = \frac{7x - 50}{3x + 91}\). We need to evaluate this function for specific values of \(x\).

Step 2: Compute \(H(1000)\)

Substitute \(x = 1000\) into the function: \(H(1000) = \frac{7 \cdot 1000 - 50}{3 \cdot 1000 + 91}\). Calculate the numerator and the denominator separately: Numerator: \(7 \cdot 1000 - 50 = 7000 - 50 = 6950\) Denominator: \(3 \cdot 1000 + 91 = 3000 + 91 = 3091\) Thus, \(H(1000) = \frac{6950}{3091} \approx 2.2476\) when rounded to four decimal places.

Step 3: Compute \(H(100,000)\)

Substitute \(x = 100,000\) into the function: \(H(100,000) = \frac{7 \cdot 100,000 - 50}{3 \cdot 100,000 + 91}\). Calculate the numerator and the denominator separately: Numerator: \(7 \cdot 100,000 - 50 = 700,000 - 50 = 699,950\) Denominator: \(3 \cdot 100,000 + 91 = 300,000 + 91 = 300,091\) Thus, \(H(100,000) = \frac{699,950}{300,091} \approx 2.3321\) when rounded to four decimal places.

Step 4: Compute \(H(1,000,000)\)

Substitute \(x = 1,000,000\) into the function: \(H(1,000,000) = \frac{7 \cdot 1,000,000 - 50}{3 \cdot 1,000,000 + 91}\). Calculate the numerator and the denominator separately: Numerator: \(7 \cdot 1,000,000 - 50 = 7,000,000 - 50 = 6,999,950\) Denominator: \(3 \cdot 1,000,000 + 91 = 3,000,000 + 91 = 3,000,091\) Thus, \(H(1,000,000) = \frac{6,999,950}{3,000,091} \approx 2.3333\) when rounded to four decimal places.

Step 5: Compute \(H(10,000,000)\)

Substitute \(x = 10,000,000\) into the function: \(H(10,000,000) = \frac{7 \cdot 10,000,000 - 50}{3 \cdot 10,000,000 + 91}\). Calculate the numerator and the denominator separately: Numerator: \(7 \cdot 10,000,000 - 50 = 70,000,000 - 50 = 69,999,950\) Denominator: \(3 \cdot 10,000,000 + 91 = 30,000,000 + 91 = 30,000,091\) Thus, \(H(10,000,000) = \frac{69,999,950}{30,000,091} \approx 2.3333\) when rounded to four decimal places.

Step 6: Conclusion

As \(x\) increases, the value of \(H(x)\) approaches \(\frac{7}{3} \approx 2.3333\). This indicates that the function \(H(x)\) approaches a horizontal asymptote of \(\frac{7}{3}\).

Key Concepts

Calculating with Rational Functions

Rational functions are quotients of two polynomials. The given function is \(H(x) = \frac{7x - 50}{3x + 91}\). When calculating values of rational functions, you substitute the given values into the function and solve for the resulting expressions. For example, to find \(H(1000)\):

• Substitute \(x = 1000\) into the function:

\(H(1000) = \frac{7 \cdot 1000 - 50}{3 \cdot 1000 + 91}\)
Next, calculate the numerator and the denominator separately:

• Numerator: \(7 \cdot 1000 - 50 = 6950\)
• Denominator: \(3 \cdot 1000 + 91 = 3091\)

Thus, \(H(1000) = \frac{6950}{3091} \approx 2.2476\) rounded to four decimal places. This process is repeated for each specified value of \(x\) in the exercise.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input grows very large or very small. In rational functions, understanding asymptotic behavior helps anticipate the value the function will approach but not reach.
For instance, calculating \(H(x)\) for large values such as \(100,000\), \(1,000,000\), and \(10,000,000\), we observe:

• The function is approaching a horizontal value of approximately \(2.3333\).

This approach indicates that as \(x\) gets larger, \(H(x)\) nears a certain constant value, known as the horizontal asymptote of the function. The asymptote helps us understand the long-term trends of the rational function.

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a function approaches but never touches as \(x\) tends to infinity or negative infinity. For the function \(H(x) = \frac{7x - 50}{3x + 91}\), we noticed that the values of \(H(x)\) for large \(x\) approach \(\frac{7}{3}\) as seen when calculating:

• For \(x = 1,000,000\), \(H(x) \approx 2.3333\)
• For \(x = 10,000,000\), \(H(x) \approx 2.3333\)

Thus, the horizontal asymptote here is \(\frac{7}{3}\), meaning that as \(x\) gets infinitely large, the function value of \(H(x)\) gets closer and closer to \(\frac{7}{3}\). Horizontal asymptotes indicate the ultimate behavior of the rational function's values.

Limits in Calculus

Limits in calculus help us understand the behavior of functions as \(x\) approaches a certain value. For rational functions like \(H(x) = \frac{7x - 50}{3x + 91}\), we use limits to determine the function's value as \(x\) approaches infinity.
We calculate the limit of \(H(x)\) as \(x \to \infty\):
\[\lim_{{x \to \infty}} \frac{7x - 50}{3x + 91}\]
Since the degrees of the numerator and the denominator are the same, we divide the coefficients of the highest degree terms:
\[\frac{7}{3}\]
Therefore, we conclude that \(\lim_{{x \to \infty}} H(x) = \frac{7}{3}\). This shows that as \(x\) grows larger without bound, the function \(H(x)\) approaches \(\frac{7}{3}\). This is essential in determining the horizontal asymptote and understanding the function's long-term behavior.