Problem 113 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(f(x)=\frac{x}{|x|}\). \\{ns. \(\left.[-1] \cup[1]\right\\}\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x}{|x|}\) is equal to 1 for positive x values (excluding 1) and -1 for negative x values (excluding -1). The function is not defined for x = 0. According to the given notation, the function remains constant in the intervals \((-\infty, -1)\cup(-1, 0)\) with a value of -1, and \((0,1)\cup(1, \infty)\) with a value of 1.

Step 1: Understanding the function's behavior for different values of x

Analyze how the function behaves for different values of x. In this case, we have two possibilities: x is positive or x is negative. For x > 0, \(\displaystyle |x|=x\), so: \(f(x)=\frac{x}{x}=1\) Now, let's analyze the case where x < 0. Here, \(\displaystyle |x|=-x\), so: \(f(x)=\frac{x}{-x}=-1\) For x = 0, the function is not defined because it leads to division by zero. With this, we have a general idea of the function's behavior.

Step 2: Domain of the Function

The domain is all the possible input values of x and in this case, the only restriction for x is when the denominator is zero. To find the domain, let's analyze the denominator: \(\displaystyle |x|\neq0\) This inequality indicates that x can be any real number except 0. Domain: \(x\in\mathbb{R}\setminus\\{0\\}\)

Step 3: Considering the Notation

Include the special notation given, \[\displaystyle \\{ns. \left[-1] \cup[1]\right\\}\] to the problem. It asks us to exclude the intervals \([-1]\) and \([1]\) from the problem. For x in the interval \((-\infty, -1)\cup(-1, 0)\), f(x)=-1. For x in the interval \((0,1)\cup(1, \infty)\), f(x)=1. So the function remains constant in those intervals.

Step 4: Conclusion

The function \(f(x) = \cfrac{x}{|x|}\) behaves differently depending on the input values of x. For positive x values (excluding 1), the function equals 1, and for negative x values (excluding -1), the function equals -1. The function is not defined for x = 0. Taking into account the special notation, the function remains constant in the intervals \((-\infty, -1)\cup(-1, 0)\) and \((0,1)\cup(1, \infty)\), with respective values of -1 and 1.

Key Concepts

Domain of a function

The domain of a function is the complete set of possible values of the independent variable, often represented as "x". In simpler terms, the domain includes all the values you can plug into the function without causing any mathematical problems, such as division by zero or taking the square root of a negative number.

For the function given in the original exercise, \(f(x) = \frac{x}{|x|}\), identifying the domain involves looking for values of \(x\) that don't cause the denominator, \(|x|\), to be zero. The absolute value \(|x|\) is zero only when \(x = 0\). This means that \(x = 0\) is excluded from the domain, as division by zero is undefined.

Therefore, the domain of this function includes all real numbers except zero. We can express this in set notation as \(x \in \mathbb{R}\setminus\{0\}\), where \(\mathbb{R}\) denotes the set of all real numbers. In interval notation, the domain is written as \((-fty, 0)\cup(0, \infty)\), emphasizing that zero is not included. This notation helps us understand where our function can be evaluated.

Absolute value

The absolute value of a number, denoted \(|x|\), represents the distance of that number from zero on the number line without considering direction. In essence, it turns negative numbers into positives and leaves positive numbers as they are. Let's break it down with examples:

• If \(x\) is positive or zero, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).

Understanding absolute value is crucial in piecewise functions, like the one in this exercise. For instance, looking at \(f(x) = \frac{x}{|x|}\), it behaves differently based on the values of \(x\).

- For \(x > 0\), \(|x| = x\), and thus, \(f(x) = \frac{x}{x} = 1\).
- For \(x < 0\), \(|x| = -x\), and thus, \(f(x) = \frac{x}{-x} = -1\).

The absolute value function helps us define the behavior of \(f(x)\) across different sections of the number line, ensuring the function responds accurately to negative and positive inputs. This behavior, combined with the specific exclusions highlighted in the exercise, defines how the function acts over its domain.

Interval notation

Interval notation is a method of representing subsets of real numbers with brackets and parentheses, providing a clear, concise way of specifying ranges.

- Use round brackets, \(( \) and \( )\), to denote that endpoints are not included in an interval.
- Use square brackets, \([ \) and \( ]\), to indicate that endpoints are included.

In the context of the given problem, interval notation helps express the domain and range exclusions in a reader-friendly format.

For example, the function \(f(x) = \frac{x}{|x|}\) is defined on \(( \-\infty, 0)\cup(0, \infty)\), which tells us the function is defined for all real numbers except zero. Special notation in the exercise suggested exclusions at further intervals, namely \([-1]\) and \([1]\), indicating that the defined values pertain to intervals excluding those nested integers:

• \((-\infty, -1)\cup(-1, 0)\)
• \((0, 1)\cup(1, \infty)\)

By using parentheses, it clarifies that the points \(-1\) and \(1\) are not part of the intervals where the function is constant. Interval notation thus provides a precise, standardized way to represent complex subsets of the number line.