Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the function is even, odd, or neither. (a) \(f(x)=|x|+3\) (b) \(g(x)=-2|x|\)

Step-by-Step Solution

Answer

The function \(f(x)=|x|+3\) is even, and the function \(g(x)=-2|x|\) is also even.

Step 1: Investigate Function f(x)

Firstly, let's focus on the function f(x) which is \(f(x)=|x|+3\). The function would look the same for x and -x as |x| gives the same result on taking negative. Hence, check by inputting -x in the function: \[f(-x) = |-x| + 3\] Since the absolute value of a number is always non-negative, we can write that |-x| = |x|. Thus, the function \(f(-x)\) yields the same output as \(f(x)\), hence it's an even function.

Step 2: Investigate Function g(x)

Now, let's look at the function \(g(x)= -2|x|\). First replace \(x\) in the function with \(-x\): \[g(-x) = -2|-x|\]Similarly, as in Step 1, the absolute value of a number is always non-negative, so we can write that |-x| = |x|. Hence, we get \(g(-x) = -2|x|\), which is the same as \(g(x)\). Therefore, g(x) is also an even function.

Key Concepts

Absolute Value

The absolute value is a fundamental concept often encountered in mathematics. It is denoted by the symbol \(|x|\), which represents the distance of a number \(x\) from zero on the number line. This distance is always non-negative. Thus, the absolute value of both positive and negative numbers yields a positive result. For instance, \(|-3| = 3\) and \(|3| = 3\).

When dealing with absolute values in functions, as we see in the exercise, replacing \(x\) with \(-x\) does not change the result. This property is essential to determining the symmetry of certain functions. In problem analysis, functions involving absolute values tend to exhibit even symmetry because \(|-x| = |x|\).

In summary, absolute values simplify the process of identifying the nature of functions in terms of evenness or oddness, especially when symmetry is a key element.

Function Analysis

Function analysis involves examining the properties and behavior of a function under various circumstances. When a function is given, our task is to explore its attributes such as continuity, symmetry, and limits. In particular, one common activity is to determine if a function is even, odd, or neither.

In the provided exercise, our main focus was assessing the symmetry of functions \(f(x) = |x| + 3\) and \(g(x) = -2|x|\), involving replacing \(x\) with \(-x\) and checking if the form remains unchanged. This approach helps identify the symmetry type, revealing deeper insights into the function’s nature and constraints.

Each type of symmetry provides specific results:

• **Even Functions**: Display identical behavior at inputs \(x\) and \(-x\).
• **Odd Functions**: Result in output that negates when inputs are negated.
• **Neither**: Do not exhibit even or odd symmetry.

Function analysis, hence, is essential for understanding and predicting the behavior of mathematical expressions under various conditions.

Even Function

An even function is characterized by its symmetric property about the y-axis. This means that for any given \(x\), \(f(x) = f(-x)\). In simple terms, plugging in \(x\) or \(-x\) into an even function yields the same result.

In the exercises provided, both \(f(x) = |x| + 3\) and \(g(x) = -2|x|\) were identified as even functions because their expressions didn’t change when \(-x\) was substituted for \(x\). This test confirmed that \(f(-x) = f(x)\) and \(g(-x) = g(x)\).

Recognizing whether a function is even is crucial in math, offering insights into graph symmetry and aiding in simplifying calculations across various scenarios, including integration and solving differential equations. Moreover, even functions simplify real-world applications, ensuring balanced responses in physical systems.