Problem 24 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The sum of the functions is \((g+f)(x) = 7x\), with domain \((-\infty, \infty)\).

Step 1: Understand the Functions

You are given two functions: \( f(x) = 3x \) and \( g(x) = 4x \). These are linear functions, where \( f(x) \) has a slope of 3 and \( g(x) \) has a slope of 4.

Step 2: Define the Domain of Each Function

The domain of a linear function like \( f(x) = 3x \) or \( g(x) = 4x \) is all real numbers, because a linear function has no restrictions on the input values. So, both \( f \) and \( g \) have the domain \( (-\infty, \infty) \).

Step 3: Adding the Functions g + f

When adding two functions \( g(x) \) and \( f(x) \), we define their sum \((g + f)(x)\) by adding their equations: \((g + f)(x) = g(x) + f(x) = 4x + 3x\).

Step 4: Simplify the Expression

Simplify \(4x + 3x\) to \(7x\). Therefore, \((g + f)(x) = 7x\). This is a new linear function created by adding the two original functions.

Step 5: Define the Domain for g+f

The domain for \((g + f)(x) = 7x\) is also all real numbers, \((-\infty, \infty)\), since both \( g(x) \) and \( f(x) \) individually have no restrictions.

Key Concepts

Linear Functions

Linear functions are simple mathematical expressions that describe a straight line. They take the form \( ax + b \), where \( a \) and \( b \) are constants. The coefficient \( a \) is called the slope, and it determines how steep the line is. In our exercise, the functions \( f(x) = 3x \) and \( g(x) = 4x \) represent linear functions where the lines pass through the origin, as there is no constant term \( b \) added. This form is known as a direct variation because the line runs directly through the point \( (0,0) \).
For any linear function, the graph of \( f(x) = 3x \) or \( g(x) = 4x \) is a straight line with:

• A consistent rate of change (the slope) across the graph.
• No curves or bends, making them simple to plot and interpret.

Understanding linear functions is crucial because they are foundation blocks in algebra, providing a basis for more complex functions.

Domain of Functions

The domain of a function refers to all the possible input values (\( x \) values) that the function can accept. For linear functions, such as \( f(x) = 3x \) and \( g(x) = 4x \), the domain is the set of all real numbers, denoted as \( (-\infty, \infty) \). This range implies that there are no constraints, and you can plug in any real number into the function.
Some key points about domains in linear functions:

• A linear function's line stretches indefinitely in either direction, reflecting its infinite domain.
• Unlike other types of functions, linear functions don't involve square roots, denominators, or logarithms, which often limit a domain.
• In practical scenarios, though, real-world problems might impose reasonable restrictions, such as a non-negative range for time or measurements.

Grasping the idea of function domains is vital as it helps ensure correct inputs when using or combining functions.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form. With the example of adding two linear functions, \( f(x) = 3x \) and \( g(x) = 4x \), the process becomes straightforward. When they are added together to form \( (g+f)(x) = 4x + 3x \), the like terms, which are both in terms of \( x \), are simply combined.
Here's how simplifying expressions works with linear functions:

• Combine like terms: For \( 4x + 3x \), combine the coefficients because they have the same variable part \( x \).
• The result is \( 7x \), which is a single, more manageable expression representing the new function.

Simplifying is a critical step in algebra that makes equations easier to solve and understand. It also helps in performing further operations like calculus or graphing, by reducing the complexity of mathematical expressions.