Domain Of $f(x)=\frac{6x-12}{x^2-2x}$: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of functions and their domains. Understanding the domain of a function is crucial in mathematics because it tells us the set of all possible input values (x-values) for which the function is defined. Today, we are going to specifically tackle the function $f(x) = \frac{6x-12}{x^2-2x}$. This is a rational function, and with rational functions, we need to be extra careful about values that might make the denominator zero, as division by zero is a big no-no in the math world. So, let's break it down step by step.

Understanding the Domain of a Function

In mathematics, the domain of a function is the set of all possible input values (often represented by x) for which the function will produce a valid output. Think of it as the function's permissible playground – the values where the function can happily operate without breaking any mathematical rules. For most functions, this playground is pretty vast, including almost all real numbers. However, certain types of functions have restrictions on their domains. These restrictions typically arise from two main scenarios: division by zero and taking the square root (or any even root) of a negative number. Let's consider these restrictions in more detail.

Division by Zero

The first major restriction comes from the fact that division by zero is undefined in mathematics. It's like trying to split a pizza into zero slices – it simply doesn't make sense! Therefore, if a function involves a fraction, we need to make sure that the denominator (the bottom part of the fraction) never equals zero. Any value of x that makes the denominator zero must be excluded from the domain. For example, in the function $f(x) = \frac{1}{x}$, the domain cannot include x = 0 because that would result in division by zero. Identifying these problematic values often involves solving an equation where the denominator is set equal to zero, and then excluding those solutions from the domain. This is a critical step in determining the domain of rational functions, which are functions expressed as a ratio of two polynomials.

Even Roots of Negative Numbers

The second key restriction comes into play when we deal with even roots, such as square roots, fourth roots, and so on. The reason for this restriction is that in the realm of real numbers, we cannot take the even root of a negative number. For example, the square root of -1 is not a real number; it's an imaginary number (denoted by i). Therefore, if a function involves an even root, we must ensure that the expression inside the root is non-negative (i.e., greater than or equal to zero). This often involves solving an inequality to determine the values of x that satisfy this condition. For instance, in the function $g(x) = \sqrt{x-2}$, we need to ensure that x - 2 ≥ 0, which means x ≥ 2. Thus, the domain of this function includes all real numbers greater than or equal to 2.

Understanding these restrictions is fundamental to accurately determining the domain of various functions. Now, let's apply these concepts to our specific function and find its domain.

Finding the Domain of $f(x)=\frac{6 x-12}{x^2-2 x}$

Okay, let's get our hands dirty and find the domain of our function, $f(x) = \frac{6x - 12}{x^2 - 2x}$. Remember, the golden rule for rational functions is to avoid division by zero. So, our mission is to figure out what values of x would make the denominator, $x^2 - 2x$, equal to zero. To do this, we'll set the denominator equal to zero and solve for x:

$x^2 - 2x = 0$

Now, we need to solve this quadratic equation. The easiest way to do this is by factoring. We can factor out an x from both terms:

$x(x - 2) = 0$

This gives us two possible solutions: either x = 0 or x - 2 = 0. Solving for x in the second equation, we get x = 2. So, the values that make our denominator zero are x = 0 and x = 2. These are the troublemakers we need to exclude from our domain!

What does this mean? It means that our function is perfectly happy for any value of x except 0 and 2. At these points, the function becomes undefined due to division by zero. Therefore, to define the domain properly, we need to exclude these values from the set of all real numbers.

We can express the domain in a few different ways. One common way is to use set notation. In set notation, we write the domain as the set of all x such that x is a real number and x is not equal to 0 or 2. Mathematically, this looks like:

{ x | x ∈ ℝ, x ≠ 0, x ≠ 2 }

Another popular way to express the domain is using interval notation. Interval notation uses parentheses and brackets to indicate the range of values included in the domain. Parentheses indicate that the endpoint is not included, while brackets indicate that it is. Since we are excluding 0 and 2, we'll use parentheses. The domain in interval notation is:

(-∞, 0) ∪ (0, 2) ∪ (2, ∞)

This notation means that the domain includes all real numbers from negative infinity up to 0 (but not including 0), then all numbers from 0 to 2 (but not including 0 or 2), and finally all numbers from 2 to positive infinity. The