Problem 13 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The function is \( f+g = 3x - 2 \), and its domain is all real numbers, \( \mathbb{R} \).

Step 1: Understanding the Functions

We have two functions, \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \). Our goal is to find \( (f + g)(x) \), which means we need to combine these functions based on their output values.

Step 2: Add the Functions

To find \( (f + g)(x) \), we simply add the expressions for \( f(x) \) and \( g(x) \). So, \( (f + g)(x) = f(x) + g(x) = (2x + 1) + (x - 3) \). Now, combine like terms: \( (f + g)(x) = 2x + x + 1 - 3 = 3x - 2 \).

Step 3: Determine the Domain

Both functions \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \) are linear functions and defined for all real numbers, since there are no restrictions such as division by zero or square roots of negative numbers. Thus, the domain of \( (f + g)(x) = 3x - 2 \) is also all real numbers, \( \mathbb{R} \).

Key Concepts

Domain of Functions

In the study of algebraic functions, the domain of a function is a fundamental concept. The domain refers to the complete set of possible input values (typically represented as 'x') that a function can accept without leading to any undefined situations. For simple linear functions, which take the form of a line,

• there are generally no restrictions on the domain.

That means every real number is a valid input. When determining the domain, we have to watch out for operations that can limit inputs, such as:

• division by zero
• taking the square root of a negative number
• logarithms of non-positive numbers

However, for linear functions like:

• \(f(x) = 2x + 1\)
• \(g(x) = x - 3\)

there are no such restrictions. Therefore, their domain is all real numbers, expressed as \(\mathbb{R}\). This means that you can plug any real number into these functions, and they will produce a valid real number output. This general rule applies to the sum of linear functions as well, such as \((f + g)(x) = 3x - 2\), which also has the domain \(\mathbb{R}\). Thus, when dealing with algebraic functions, identifying the domain is the first step to understanding how the function behaves.

Function Addition

Adding functions is a basic operation in algebra, where you combine two or more functions to create a new one. This process is straightforward:

• you add the outputs of the functions for the same input value.

For example, if we have:

• \(f(x) = 2x + 1\)
• \(g(x) = x - 3\)

we find their sum, \((f + g)(x)\), by adding their outputs. This results in:

• \( (f + g)(x) = (2x + 1) + (x - 3) \)

Once you line up and add the like terms, you simplify the expression:

• \(3x - 2\)

This new function, \(3x - 2\), is the result of the function addition. It's important to note that the way to "add" functions is not to combine the input variables but the output, or the values that the functions produce for those inputs. By understanding function addition, you can combine smaller, simpler functions into more complex ones, aiding in the study and application of algebraic expressions.

Linear Functions

Linear functions are among the simplest and most essential types of algebraic functions. They are broadly used in various fields due to their straightforward nature. A linear function can be recognized by its structure:

• it usually takes the form
  \(y = mx + c\), where:
• \(m\) represents the slope of the line, and
• \(c\) represents the y-intercept.

The slope, \(m\), describes how steep the line is. A higher slope value indicates a steeper incline, while a flat line has a slope of zero. The y-intercept, \(c\), indicates where the line crosses the y-axis. In the functions \(f(x) = 2x + 1\) and \(g(x) = x - 3\),

• \(f(x)\) has a slope of 2, while its y-intercept is 1.
• \(g(x)\) has a slope of 1, and its y-intercept is -3.

The characteristic feature of these functions is that their graphs produce a straight line when plotted on a coordinate plane. This linearity makes it easy to predict and analyze the behavior of these functions for given inputs. Understanding linear functions is crucial because it sets the groundwork for more complex functions encountered in algebra and calculus.