Problem 16 - Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x-3 .\) Find each function and give its domain. See Example 1. $$ g+f $$

Step-by-Step Solution

Answer

\((g+f)(x) = 3x - 2\); domain: all real numbers.

Step 1: Understand the Given Functions

We are given two functions: \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \). Our goal is to find \( (g+f)(x) \), which represents the sum of these two functions.

Step 2: Add the Functions

To find \( (g+f)(x) \), we need to add the two given functions: \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \). We add them as follows:\[(g+f)(x) = g(x) + f(x) = (x - 3) + (2x + 1)\]

Step 3: Simplify the Expression

Combine like terms in the expression:\[(g+f)(x) = x - 3 + 2x + 1\]Combine \(x\) terms and constant terms:\[(g+f)(x) = 3x - 2\]

Step 4: Determine the Domain

The functions \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \) are linear functions. Linear functions are defined for all real numbers. The domain of the sum, \( (g+f)(x) = 3x - 2 \), is also all real numbers. Thus, the domain is \( (-\infty, \infty) \).

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions in mathematics. Linear functions are described by a polynomial equation with the highest degree of 1. This means the graph of a linear function is a straight line. An example of a linear function is given by the formula:\[ f(x) = ax + b \]Here:

• \( a \) is the coefficient of \( x \), representing the slope of the line. It tells us how steep the line is.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

Let's take the example from the problem: \( f(x) = 2x + 1 \). Here, the slope \( a \) is 2 and the y-intercept \( b \) is 1. This suggests that for every 1 unit increase in \( x \), \( f(x) \) increases by 2 units. Hence, linear functions model relationships where changes in one quantity are proportional to changes in another.

Function Addition

Adding functions is a fundamental operation in algebra, symbolized by \( (g+f)(x) \). It essentially involves summing two functions to create a new, composite function. When you add two functions, like \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), you get:\[ (g+f)(x) = g(x) + f(x) = (x - 3) + (2x + 1) \]To find \( (g+f)(x) \), simply add corresponding terms together.

• Add the \( x \) terms: \( x + 2x = 3x \)
• Add the constant terms: \( -3 + 1 = -2 \)

This simplifies to:\[ (g+f)(x) = 3x - 2 \]Thus, function addition involves combining each component of the functions being summed. The result is a new function that includes the contributions from both original functions. This new function behaves differently, depending on the coefficients and constants involved.

Domain of a Function

The domain of a function refers to all the possible input values (commonly \( x \)-values) that a function can accept. For linear functions, like \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), the domain is typically all real numbers. This is because linear functions have no limitations or restrictions—there are no values of \( x \) for which the function cannot produce a valid output.When dealing with the domain of a sum of functions, such as \( (g+f)(x) = 3x - 2 \), the domain is also all real numbers. This is because each component function (e.g., \( f(x) \) and \( g(x) \) ) in the sum has a domain of all real numbers. Thus, the sum’s domain is a result of the intersection of their domains, which is still all real numbers. In mathematical notation, this is represented as:\[ (-\infty, \infty) \]Linear functions, unlike some other types of functions, typically extend infinitely in both directions along the x-axis, making them applicable to any real-world scenario where proportional relationships are observed.