Verified Answer :
- The value of \(f(9)\) is \(\frac{5}{8}\).
- The \(x\)-intercept is \((-\frac{6}{11}, 0)\).
- The \(y\)-intercept is \((0, -\frac{1}{2})\).
- The vertical asymptote is \(x = \frac{3}{5}\) and the horizontal asymptote is \(y = \frac{11}{20}\).
- Final answer: \(f(9) = \boxed{\frac{5}{8}}\), \(x\)-int: \(\boxed{(-\frac{6}{11}, 0)}\), \(y\)-int: \(\boxed{(0, -\frac{1}{2})}\), Vert. Asym: \(\boxed{x = \frac{3}{5}}\), Horiz. Asym: \(\boxed{y = \frac{11}{20}}\)
Explanation
- Understanding the Problem and Data Analysis
The function given is \(f(x) = \frac{11x + 6}{20x - 12}\). This is a rational function, which is defined for all real values of \(x\) except those where the denominator is zero. To analyze its properties, we will evaluate the function at a specific point, find its intercepts, and determine its asymptotic behavior.
- Evaluating the Function at \(x=9\)
To evaluate \(f(9)\), we substitute \(x = 9\) into the function:
\[f(9) = \frac{11(9) + 6}{20(9) - 12} = \frac{99 + 6}{180 - 12} = \frac{105}{168}.\]
Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor, 21:
\[\frac{105 \div 21}{168 \div 21} = \frac{5}{8}.\]
Thus, \(f(9) = 0.625\).
- Finding the x-intercept
The \(x\)-intercept occurs where \(f(x) = 0\). This happens when the numerator is zero and the denominator is non-zero:
\[11x + 6 = 0 \implies 11x = -6 \implies x = -\frac{6}{11}.\]
Since the denominator \(20(-\frac{6}{11}) - 12 = -\frac{120}{11} - \frac{132}{11} = -\frac{252}{11}
eq 0\), the \(x\)-intercept is at the point \((-\frac{6}{11}, 0)\).
- Finding the y-intercept
The \(y\)-intercept occurs where \(x = 0\):
\[f(0) = \frac{11(0) + 6}{20(0) - 12} = \frac{6}{-12} = -\frac{1}{2}.\]
Thus, the \(y\)-intercept is at the point \((0, -0.5)\).
- Finding the Vertical Asymptote
The vertical asymptote occurs where the denominator is zero (and the numerator is non-zero):
\[20x - 12 = 0 \implies 20x = 12 \implies x = \frac{12}{20} = \frac{3}{5}.\]
So, the equation of the vertical asymptote is \(x = 0.6\).
- Finding the Horizontal Asymptote
For a rational function of the form \(f(x) = \frac{ax + b}{cx + d}\), the horizontal asymptote is the ratio of the leading coefficients, \(y = \frac{a}{c}\):
\[y = \frac{11}{20} = 0.55.\]
So, the equation of the horizontal asymptote is \(y = 0.55\).
Examples
Rational functions are used in many real-world scenarios, such as modeling cost-benefit ratios or concentration levels in chemistry. For example, if you are modeling the concentration of a chemical in a tank as water flows in and out, the concentration might be represented by a function similar to \(f(x) = \frac{11x + 6}{20x - 12}\), where \(x\) represents time. Understanding asymptotes helps predict the long-term behavior (horizontal asymptote) and potential points where the model breaks down (vertical asymptote, such as when the tank is empty).