- The domain of a rational function includes all real numbers except those where the denominator equals zero.
- For \(f(x) = \frac{x-4}{2x^2 - 5x - 12}\), solve the denominator \(2x^2 - 5x - 12 = 0\) to find the points where the function is undefined.
- The quadratic formula gives the roots \(x = -\frac{3}{2}\) and \(x = 4\).
- The domain is all real numbers except these two points, written as \(\boxed{(-\infty, -\frac{3}{2}) \cup (-\frac{3}{2}, 4) \cup (4, \infty)}\).
Explanation
- Analyzing the Function and Identifying the Domain Condition
The domain of a rational function consists of all real numbers except those that cause the denominator to be equal to zero. For the given function \(f(x) = \frac{x-4}{2x^2 - 5x - 12}\), the denominator is \(2x^2 - 5x - 12\). We must identify all values of \(x\) such that \(2x^2 - 5x - 12 = 0\) and exclude these from the real numbers.
- Solving for Excluded Values
To find the values to exclude, we solve the quadratic equation \(2x^2 - 5x - 12 = 0\). We can use the quadratic formula, where \(a = 2\), \(b = -5\), and \(c = -12\). The discriminant is given by \(D = b^2 - 4ac = (-5)^2 - 4(2)(-12) = 25 + 96 = 121\). The roots are then \(x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{121}}{2(2)} = \frac{5 \pm 11}{4}\).
This gives us two solutions:
- \(x_1 = \frac{5 - 11}{4} = \frac{-6}{4} = -\frac{3}{2}\)
- \(x_2 = \frac{5 + 11}{4} = \frac{16}{4} = 4\)
Thus, the function is undefined at \(x = -\frac{3}{2}\) and \(x = 4\).
- Determining the Domain in Interval Notation
The domain of the function includes all real numbers except for \(x = -\frac{3}{2}\) and \(x = 4\). In interval notation, we represent this as the set of all real numbers excluding those two points, which is \((-\infty, -\frac{3}{2}) \cup (-\frac{3}{2}, 4) \cup (4, \infty)\). Comparing this to the provided options, this corresponds to the second option.
Examples
Understanding domains of functions is crucial in fields like physics and engineering, where mathematical models describe real-world phenomena. For example, if a function represents the speed of a vehicle over time, the domain identifies valid time intervals where the model is applicable. Identifying points where a function is undefined helps avoid critical errors in calculations, such as division by zero, which could lead to nonsensical results in physical simulations or control systems. Correctly defining the domain ensures your models are robust and reliable for predicting behavior in any practical application.