Problem 24 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The sum is \((g+f)(x) = 7x\). The domain is all real numbers \(\mathbb{R}\).

Step 1: Understand the Problem

We need to find the sum of two functions, \(f(x)\) and \(g(x)\), where \(f(x) = 3x\) and \(g(x) = 4x\). We will also identify the domain of the sum of these functions.

Step 2: Write Down the Expressions for Each Function

We know \(f(x) = 3x\) and \(g(x) = 4x\).

Step 3: Add the Functions

Determine the sum \((g+f)(x)\) by adding the expressions of \(f(x)\) and \(g(x)\). The sum is: \[(g+f)(x) = g(x) + f(x) = 4x + 3x\]

Step 4: Simplify the Expression

Combine like terms to simplify the expression from the addition: \[(g+f)(x) = 7x\]

Step 5: Determine the Domain of \((g+f)\)

Both functions \(f(x) = 3x\) and \(g(x) = 4x\) are linear functions, which are defined for all real numbers. Thus, the domain of \((g+f)(x) = 7x\) is also all real numbers, denoted by \(\mathbb{R}\).

Key Concepts

Function Addition

Adding functions in algebra involves combining their outputs for the same input value. This process is simple and just like adding numbers together. Let's say you have two functions, like the ones in the exercise:

• \( f(x) = 3x \)
• \( g(x) = 4x \)

To find the function \((g+f)(x)\), you simply add them:

• \((g+f)(x) = g(x) + f(x) = 4x + 3x\).

When adding these functions, you gather all like terms together, the result here is \(7x\).
This means for any value of \(x\), the output of \((g+f)(x)\) is seven times that value. It's a straightforward process that becomes even more intuitive with practice.

Linear Functions

Linear functions are a type of function that creates a straight line when graphed. Simplified, a linear function can be written in the form \(f(x) = mx + b\).

• \(m\) is the slope of the line, indicating its steepness.
• \(b\) represents the y-intercept, where the line crosses the y-axis.

In the given exercise:

• \(f(x) = 3x\) and \(g(x) = 4x\)

The slope for \(f(x)\) is 3, and for \(g(x)\), it is 4. Notice they both don't have a y-intercept \(b\) term so they cross the origin (0,0) in a graph.
Linear functions like these extend infinitely in both directions across the x-axis and y-axis, making them simple yet important in understanding more complex algebraic concepts.

Domain of Functions

The domain of a function is crucial as it tells us all the possible input values that the function accepts. For linear functions, such as \(f(x) = 3x\) and \(g(x) = 4x\), their domains are straightforward. Linear functions are defined for all real numbers because there's no division by zero or square roots of negative numbers involved, which are common restrictions in other types of functions.

• For \(f(x) = 3x\), the domain is all real numbers: denoted as \(\mathbb{R}\).
• Similarly, \(g(x) = 4x\) also has a domain of all real numbers: \(\mathbb{R}\).

Thus, the combined function \((g+f)(x) = 7x\) will also have a domain of all real numbers. Knowing the domain ensures accuracy when calculating or graphing functions, as it specifies the full set of inputs the function can handle without errors.