Problem 23 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(f(x)=x\) and \(g(x)=|x|\), then what is \((f+g)\) \((x)\) equal to? \(\quad\) INDA-2008] (a) 0 for all \(x \in R\) (b) \(2 x\) for all \(x \in R\) (c) \(\left\\{\begin{array}{c}2 x \text { for } x \geq 0 \\ 0 \text { for } x<0\end{array}\right.\) (d) \(\left\\{\begin{array}{c}0 \text { for } x \geq 0 \\ 2 x \text { for } x<0\end{array}\right.\)

Step-by-Step Solution

Answer

The answer is (c): \(2x\) for \(x \geq 0\) and \(0\) for \(x < 0\).

Step 1: Understanding the Functions

Identify the functions given in the problem. Here, \( f(x) = x \) and \( g(x) = |x| \). \( f(x) \) is the identity function, which outputs \( x \) for any input \( x \). \( g(x) \) is the absolute value function, which outputs the non-negative value of \( x \).

Step 2: Define (f + g)(x)

To find \((f+g)(x)\), we need to add the functions \( f(x) \) and \( g(x) \). Thus, \((f+g)(x) = f(x) + g(x) = x + |x|\).

Step 3: Simplify (f + g)(x) in Different Cases

Consider the expression \(x + |x|\) in two scenarios: when \(x \geq 0 \) and when \(x < 0\). For \(x \geq 0\), \(|x| = x\), so \((f+g)(x) = x + x = 2x\). For \(x < 0\), \(|x| = -x\), so \((f+g)(x) = x + (-x) = 0\).

Step 4: Final Answer

Combine the results into a piecewise function. For \(x \geq 0\), \((f+g)(x) = 2x\), and for \(x < 0\), \((f+g)(x) = 0\). Therefore, \((f+g)(x)\) is: \[\begin{cases} 2x & \text{if } x \geq 0 \ 0 & \text{if } x < 0 \end{cases}\]

Key Concepts

Identity Function

The identity function is a fundamental building block in mathematics. It is defined as \( f(x) = x \). Essentially, this function maps any input \( x \) to the same value \( x \). For every input, whatever goes into the function, comes out unchanged. This property makes the identity function incredibly easy to understand and useful in algebra and calculus.
It helps in simplifying expressions and solving equations, often acting as a neutral element in algebraic manipulations. When combined with other functions, it doesn't alter them but retains the distinct characteristics of the other function.

Absolute Value Function

The absolute value function, denoted as \( g(x) = |x| \), is another essential mathematical concept. This function outputs the non-negative value of a number. It essentially measures the "magnitude" of a number, ignoring any negative signs. For instance:

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

This function is critical in piecewise definitions because it behaves differently across the number line. It's essential in real-world applications, such as calculating distances, because distance is always a non-negative quantity. Understanding how this function operates on different ranges of numbers is key to solving problems involving piecewise functions.

Function Addition

Function addition, represented as \((f+g)(x)\), involves adding two functions, both symbolically and numerically. In this context, adding functions means computing \(f(x) + g(x)\). Each of the functions is evaluated at a certain \(x\), and their results are added together.
Consider the functions \( f(x) = x \) and \( g(x) = |x| \). The operation \((f+g)(x) = x + |x|\) combines their outputs. Depending on the value of \(x\), the sum will differ:

• For \(x \geq 0\), this becomes \( x + x = 2x \).
• For \(x < 0\), it simplifies to \( x + (-x) = 0 \).

By understanding function addition, we can effectively create new functions that exhibit different properties based on the components involved.

Algebraic Expressions

Algebraic expressions involve combinations of variables and constants through operations such as addition, subtraction, and multiplication. In the context of functions, an expression like \( x + |x| \) is considered algebraic. Handling these expressions requires an understanding of basic algebraic manipulations and properties of the involved components.
In particular, piecewise functions like this one involve using logical conditions to handle different cases within the algebraic expression. The goal is to evaluate the expression accurately across different domains of \( x \). This understanding is pivotal to solving for the function's output in various scenarios, ensuring algebraic expressions produce the intended results based on the input conditions.