Problem 1 - Intermediate Algebra

Question

Fill in the blanks. The _____ of \(f\) and \(g,\) denoted as \(f+g,\) is defined by \((f+g)(x)=\square\) and the _____ of \(f\) and \(g\), denoted as \(f-g,\) is defined by \((f-g)(x)=\square\).

Step-by-Step Solution

Answer

The sum of two functions \((f+g)(x)\) is defined as \(f(x) + g(x)\), and the difference \((f-g)(x)\) is defined as \(f(x) - g(x)\).

Step 1: Understand Sum of Functions

The sum of two functions, say \( f \) and \( g \), is simply another function that is obtained by adding the values of \( f \) and \( g \) at each point \( x \). The notation for the sum of two functions is \( f + g \). The sum evaluated at any \( x \) is given as \((f + g)(x) = f(x) + g(x)\).

Step 2: Apply Sum Definition

Fill in the first blank with 'sum', as it describes the operation of adding two functions. For the equation, fill in the box with \(f(x) + g(x)\) since the sum of the functions at \( x \) is obtained by adding \( f(x) \) and \( g(x) \).

Step 3: Understand Difference of Functions

The difference of two functions \( f \) and \( g \) is another function, obtained by subtracting the value of \( g \) from the value of \( f \) at each point \( x \). The difference is denoted as \( f - g \). The expression \((f - g)(x) = f(x) - g(x)\) shows the difference of the functions evaluated at \( x \).

Step 4: Apply Difference Definition

In the sentence, fill in the second blank with 'difference' as it refers to the operation of subtracting the functions. In the equation, fill in the box with \(f(x) - g(x)\) as this represents the difference when \( g(x) \) is subtracted from \( f(x) \).

Key Concepts

Sum of Functions

When studying functions, the concept of adding two functions together is useful. It helps in analyzing and simplifying various problems. The result of adding two functions is called the "sum of functions." If you have two functions, denoted as \( f(x) \) and \( g(x) \), their sum is represented by \( f+g \). Mathematically, this is expressed as:

• \((f+g)(x) = f(x) + g(x)\)

Here, \((f+g)(x)\) means that for each point \(x\), you add the outputs of \(f\) and \(g\). For example, if \(f(2) = 3\) and \(g(2) = 5\), then \((f+g)(2) = f(2) + g(2) = 3 + 5 = 8\). Imagine it like stacking two sets of numbers on top of each other and adding them neatly. In applications, the sum of functions is crucial in physics, engineering, and economics where combined effects of multiple variables or processes need to be analyzed. Understanding the sum of functions allows you to break complex problems into manageable parts and analyze each contribution separately.

Difference of Functions

The difference of functions is equally important as the sum. It involves subtracting one function from another point by point. If you have functions \( f(x) \) and \( g(x) \), the difference is expressed as \( f-g \). The mathematical representation is:

• \((f-g)(x) = f(x) - g(x)\)

With this equation, for each value of \(x\), you calculate the result of \(f(x)\) minus \(g(x)\). For instance, if \(f(4) = 7\) and \(g(4) = 3\), then \((f-g)(4) = f(4) - g(4) = 7 - 3 = 4\). Think of it like peeling away layers: you remove the value of \(g(x)\) from \(f(x)\) at every point.The concept of the difference of functions is key in scenarios where distinctions among processes or data sets need to be parsed out, such as finding profit by subtracting costs from revenue, or removing noise from signals in technology. By understanding this difference, you can isolate changes and focus on specific parts of a larger problem.

Function Notation

Function notation is a convenient way of displaying information about a function and how it interacts with different inputs. It utilizes a simple yet powerful way to express which operations are being performed on functions. When you see \( f(x) \), it's telling you about a function named \( f \) and that \( x \) is the input. Similarly, for two functions \( f \) and \( g \), expressions like \( f+g \) or \( f-g \) convey operations involving both functions.By using function notation, mathematicians and students can easily and concisely represent operations on functions for clear communication. Here's how it helps:

• Consistent usage makes complex equations easier to read.
• Allows for straightforward substitution of values into expressions, ensuring efficient calculations.
• Encourages understanding of relationships between variables and functions.

The notation plays a significant role from high school algebra to higher-level calculus and beyond, where precision and clarity are paramount. It is essential to become familiar with this notation as it simplifies working with multiple functions and the operations performed on them.