Verified Answer :
- Evaluate \(f(6)\) by substituting 6 into the function, yielding \(\frac{47}{40}\) or \(1.175\).
- Determine the \(x\)-intercept by setting the numerator to zero, giving the point \((-\frac{5}{7}, 0)\).
- Find the \(y\)-intercept by calculating \(f(0)\), resulting in the point \((0, -0.25)\).
- Identify the vertical asymptote where the denominator is zero, yielding \(x = 2\), and find the horizontal asymptote as the ratio of leading coefficients, yielding \(y = 0.7\).
- Summary of results: \(f(6) = \boxed{1.175}\), \(x\)-intercept = \(\boxed{(-5/7, 0)}\), \(y\)-intercept = \(\boxed{(0, -0.25)}\), VA = \(\boxed{x = 2}\), HA = \(\boxed{y = 0.7}\).
Explanation
- Analysis of the Rational Function
The function given is \(f(x) = \frac{7x + 5}{10x - 20}\). This is a rational function, which is defined for all real numbers except where the denominator is zero. To analyze this function, we will calculate specific points and identify key graphical features like intercepts and asymptotes.
- Evaluating f(6)
To evaluate \(f(6)\), we substitute \(x = 6\) into the function expression:
\[f(6) = \frac{7(6) + 5}{10(6) - 20} = \frac{42 + 5}{60 - 20} = \frac{47}{40}.\]
As a decimal, \(\frac{47}{40} = 1.175\).
- Finding the x-intercept
The \(x\)-intercept occurs when the function value is zero, i.e., \(f(x) = 0\). This requires the numerator to be zero while the denominator remains non-zero:
\[7x + 5 = 0 \implies 7x = -5 \implies x = -\frac{5}{7}.\]
Since the denominator at \(x = -\frac{5}{7}\) is \(10(-\frac{5}{7}) - 20
eq 0\), the \(x\)-intercept is at the point \((-\frac{5}{7}, 0)\).
- Finding the y-intercept
The \(y\)-intercept occurs when \(x = 0\). Substituting this into the function:
\[f(0) = \frac{7(0) + 5}{10(0) - 20} = \frac{5}{-20} = -\frac{1}{4}.\]
Thus, the \(y\)-intercept is at the point \((0, -0.25)\).
- Finding Asymptotes and Conclusion
The vertical asymptote occurs where the denominator is zero and the numerator is non-zero:
\[10x - 20 = 0 \implies 10x = 20 \implies x = 2.\]
At \(x = 2\), the numerator is \(7(2) + 5 = 19
eq 0\), so the equation of the vertical asymptote is \(x = 2\). The horizontal asymptote is found by examining the limit of \(f(x)\) as \(x\) approaches \(\pm \infty\). By comparing the leading coefficients of the numerator and denominator:
\[\lim_{x \to \infty} \frac{7x + 5}{10x - 20} = \frac{7}{10} = 0.7.\]
Therefore, the horizontal asymptote is \(y = 0.7\).
Examples
Rational functions are used in many fields, such as economics, to model cost functions. For example, if a factory has a fixed cost for machinery and a variable cost per item produced, the average cost per item can be modeled as a rational function. Understanding how to find asymptotes helps engineers and economists determine long-term trends—the horizontal asymptote tells them the cost to produce an item in a very large-scale operation, while the vertical asymptote might highlight a production limit or a point where the model breaks down.