Problem 21 - Intermediate Algebra

Question

Let \(f(x)=3 x\) and \(g(x)=4 x .\) Find each function and give its domain. See Example 1. $$ f+g $$

Step-by-Step Solution

Answer

\((f+g)(x) = 7x\); domain is all real numbers.

Step 1: Understanding the Functions

We are given two functions: \( f(x) = 3x \) and \( g(x) = 4x \). We need to find the sum of these functions, noted as \( (f + g)(x) \).

Step 2: Adding the Functions

When we add two functions, we simply add their expressions. So, \((f + g)(x) = f(x) + g(x) = 3x + 4x\).

Step 3: Simplifying the Expression

Combine like terms: \(3x + 4x = 7x\). Thus, \((f + g)(x) = 7x\).

Step 4: Determining the Domain

Both \( f(x) \) and \( g(x) \) are linear functions, which means their domain is all real numbers. Since the operation performed (addition) does not restrict this domain, the domain of \((f + g)(x)\) is also all real numbers: \( (-\infty, \infty) \).

Key Concepts

Linear Functions

Linear functions are a fundamental type of function in mathematics, often represented by the formula \(f(x) = ax + b\). These functions graph as straight lines, where the coefficient \(a\) is the slope of the line, and \(b\) is the y-intercept. The slope \(a\) indicates how steep the line is and in which direction it tilts. A positive slope creates an upward slant from left to right, while a negative slope causes a downward slant.
Linear functions are straightforward due to their constant rate of change. This means, no matter where you are on the line, the change in y for a given change in x is always the same. In our exercise, both functions \(f(x) = 3x\) and \(g(x) = 4x\) lack a constant (\(b = 0\)), meaning they pass through the origin (0,0), further simplifying their graph and behavior.

Function Addition

Function addition involves combining two or more functions into a single expression. The operation is fairly simple: you add the outputs of the functions for the same input. Mathematically, for two functions \(f(x)\) and \(g(x)\), their sum \((f + g)(x)\) is given by the formula \(f(x) + g(x)\).
In our example, adding the functions \(f(x) = 3x\) and \(g(x) = 4x\) gives: \((f + g)(x) = 3x + 4x\). We combine like terms, resulting in \((f + g)(x) = 7x\).
This operation is particularly useful when you need to analyze the combined effect of two different processes represented by linear functions, allowing you to maintain an understanding of their cumulative impact.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable, typically represented as \(x\), that you can input into the function without causing issues like division by zero or taking square roots of negative numbers.
Linear functions, like the examples \(f(x) = 3x\) and \(g(x) = 4x\), have a domain of all real numbers, denoted as \((-\infty, \infty)\). This is because there are no restrictions like those found in radical or rational functions.
When performing operations like addition on functions, it’s essential to consider the domains. Since both \(f(x)\) and \(g(x)\) have domains of all real numbers, \((f + g)(x)\) also retains this domain, allowing any real number to be substituted for \(x\) without any issue.