Problem 55 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) for which each function is continuous. \(f(x)=|x+1|\)

Step-by-Step Solution

Answer

The function \(f(x) = |x + 1|\) is continuous for all real values of x, as the modulus of a continuous linear function, in this case, \(u(x) = x + 1\), is also continuous for all real x.

Step 1: Identify the modulus function inside f(x)

We see that the given function is the modulus function with argument (x+1).

Step 2: State continuity of modulus functions

We know that the modulus function |u(x)|, where u(x) is any real-valued function, is continuous for all values of x when the function inside the modulus, u(x), is continuous i.e. if u(x) is continuous, so is |u(x)|. Now, the function inside the modulus is the linear function: \(u(x) = x+1\)

Step 3: Identify the continuity of u(x)

As a linear function, u(x) = x+1 is continuous for all real values of x.

Step 4: Conclude continuity of f(x) for all x values

Since u(x) = x+1 is continuous for all real values of x, the modulus function, |u(x)|, should also be continuous for all real values of x. Therefore, the function \(f(x) = |x + 1|\) is continuous for all real values of x.

Key Concepts

Understanding the Modulus Function

The modulus function is a fundamental concept in mathematics. It is commonly represented as \[ f(x) = |x| \] This function takes any input value and returns its absolute value, which is always a non-negative number. In simpler terms, the modulus function measures the distance of a number from zero, without considering the direction along the number line.
For instance:

• |3| equals 3, since 3 is already positive.
• |-3| equals 3, because distance cannot be negative.

The modulus function impacts continuity significantly because it incorporates absolute values, which are smooth and unbroken across their domain. When you have a function like \( f(x) = |x + 1| \), it means the function is always taking the absolute value of \( x + 1 \), ensuring continuity across all real numbers.

Exploring Linear Functions

Linear functions are among the simplest types of functions in mathematics. They have the form:\[ u(x) = mx + b \] where "m" represents the slope and "b" is the y-intercept. A key aspect of linear functions is that they have a constant rate of change, depicted by straight lines when graphed. Important characteristics of linear functions include:

• They are defined at all points on the real number line, making them continuous everywhere.
• The graph of a linear function is a straight line with no breaks or jumps.

In our exercise, the linear expression \( u(x) = x + 1 \) plays a crucial role in confirming continuity, as it is continuous for all values of \( x \). When combined with the modulus function, it guarantees that the resulting function \( |x + 1| \) is also continuous everywhere.

Real-Valued Functions and Their Continuity

Real-valued functions are functions that take real numbers as inputs and produce real numbers as outputs. These functions are essential in both pure and applied mathematics, since they allow for easy interpretation and analysis.
Key features of real-valued functions include:

• They can be expressed in various forms such as polynomial, exponential, and trigonometric functions.
• They enable seamless plotting on the Cartesian plane, providing a visual representation of mathematical concepts.

In terms of continuity, a function being real-valued suggests it does not have points of discontinuity within its domain. For \( f(x) = |x + 1| \), since it is formed by a real-valued linear function and the modulus operation, it remains smooth and continuous over all real numbers.
Ultimately, understanding how these three concepts interconnect can simplify the analysis of any piecewise or composite function, like the one given in the exercise, ensuring you grasp the importance of continuity in various mathematical contexts.