Problem 13 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The function is \((f+g)(x) = 3x - 2\) with domain \((-\infty, \infty)\).

Step 1: Identify Functions

The problem provides us with two functions, \( f(x) = 2x+1 \) and \( g(x) = x-3 \). These functions are both linear, as they are in the form \( ax + b \).

Step 2: Define the Sum of Functions

To find \((f+g)(x)\), add the two functions together. That is \((f+g)(x) = f(x) + g(x)\).

Step 3: Perform the Addition

Substitute the given functions into the expression: \((f+g)(x) = (2x + 1) + (x - 3)\). Simplify this by combining like terms: \(2x + 1 + x - 3 = 3x - 2\).

Step 4: Identify the Domain

Since both \(f(x)\) and \(g(x)\) are linear polynomials, their domain is all real numbers. Thus, the domain of \( (f+g)(x) \) is also all real numbers \( (-\infty, \infty) \).

Key Concepts

Understanding Linear Functions

Linear functions are one of the simplest and most fundamental types of functions in mathematics. They are functions that create a straight line when graphed on the coordinate plane. Every linear function can be expressed in the form \[ f(x) = ax + b \]where:

• \(a\) is the coefficient of \(x\) and determines the slope of the line. It describes how steep the line is.
• \(b\) is the y-intercept, showing the point where the line crosses the y-axis.

In our exercise, we see that both \(f(x) = 2x + 1\) and \(g(x) = x - 3\) are linear functions.
This means they will graph as straight lines and have predictable behaviors.Linear functions are easy to work with because of their consistent slope, making them a great starting point in understanding more complex functions.

Exploring the Domain of a Function

The domain of a function refers to the complete set of all possible input values (usually \(x\) values) that the function can accept.
For a linear function, like the ones we have in the exercise, there are no restrictions on these input values.
The domain of a linear function is all real numbers, denoted as \[(-\infty, \infty)\].This essentially means that you can plug in any real number for \(x\) and it will produce a corresponding \(y\)-value output.
Having infinite possibilities for \(x\) makes linear functions very flexible and simple to analyze since there are no inherent limitations except when explicitly defined by context or problem.

Calculating the Sum of Functions

The process of finding the sum of two functions involves adding them together, much like regular addition.When provided with functions such as \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), the sum is found by:

• Writing the combined function as \((f+g)(x)\).
• Add each pair of terms from \(f(x)\) and \(g(x)\): \((f+g)(x) = (2x + 1) + (x - 3)\).
• Simplify by combining like terms: \[2x + x + 1 - 3 = 3x - 2\].

This results in a new linear function, \((f+g)(x) = 3x - 2\), which is also a linear function. Its domain remains the same as its components, covering all real numbers since linear functions have unlimited/unrestricted domains.
This operation helps to combine the effects or changes each function represents into a single, unified expression.