Find the domain of the function.
$f(x)=\frac{7-x}{x^2-4 x}$
The domain is □ .
The domain of a rational function \(f(x) = \frac{P(x)}{Q(x)}\) consists of all real numbers \(x\) for which the denominator \(Q(x)\) is not equal to zero. In this problem, the function is \(f(x) = \frac{7-x}{x^2-4x}\), so we need to identify the values of \(x\) that make the denominator \(x^2 - 4x\) equal to zero. These values must be excluded from the domain of the function.
To find the values where the denominator is zero, we set the expression \(x^2 - 4x\) equal to zero:
\[x^2 - 4x = 0\]
We can factor out \(x\) from the expression:
\[x(x - 4) = 0\]
This equation holds true if either factor is zero. Setting each factor to zero gives:
\[x = 0 \quad \text{or} \quad x - 4 = 0\]
Solving these simple linear equations, we find:
\[x = 0 \quad \text{or} \quad x = 4\]
These are the values that cause the denominator to be zero, meaning the function is undefined at these points.
Since the function is undefined at \(x=0\) and \(x=4\), we must exclude these two points from the set of all real numbers \(\mathbb{R}\). The set of all real numbers excluding \(0\) and \(4\) can be expressed using interval notation as the union of three intervals: all numbers less than 0, all numbers between 0 and 4, and all numbers greater than 4.
Thus, the domain is \((-\infty, 0) \cup (0, 4) \cup (4, \infty)\).
Understanding the domain of a function is crucial in real-world applications, such as engineering or economics. For example, if a function represents the cost of producing \(x\) units of a product, and the denominator involves an expression that is zero at a certain production level (like zero units, where you might have fixed costs but not production costs), knowing where the function is undefined helps determine the feasible range of inputs for your model. This ensures that calculations remain valid and prevents errors in predictive analysis.