Problem 86 - Elementary Algebra

Question

Calculate \((f+g)(x)\) and \((f-g)(x)\) and state the restrictions to the domain. \(f(x)=5 x+2\) and \(g(x)=3 x+4\)

Step-by-Step Solution

Answer

(f+g)(x) = 8x + 6, (f-g)(x) = 2x - 2. No domain restrictions.

Step 1: Define the Expression for (f+g)(x)

To find \((f+g)(x)\), add the functions \(f(x)\) and \(g(x)\) together: \((f+g)(x) = f(x) + g(x)\). Using the given functions, substitute: \((f+g)(x) = (5x + 2) + (3x + 4)\).

Step 2: Simplify (f+g)(x)

Combine like terms in the expression: \((f+g)(x) = 5x + 3x + 2 + 4 = 8x + 6\). Therefore, \((f+g)(x) = 8x + 6\).

Step 3: Define the Expression for (f-g)(x)

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\): \((f-g)(x) = f(x) - g(x)\). Using the given functions, substitute: \((f-g)(x) = (5x + 2) - (3x + 4)\).

Step 4: Simplify (f-g)(x)

Distribute the negative sign and combine like terms: \((f-g)(x) = 5x - 3x + 2 - 4 = 2x - 2\). Therefore, \((f-g)(x) = 2x - 2\).

Step 5: Determine the Domain Restrictions

Both \(f(x) = 5x + 2\) and \(g(x) = 3x + 4\) are linear functions, which are defined for all real numbers. Thus, \((f+g)(x)\) and \((f-g)(x)\) are also defined for all real numbers. There are no restrictions to the domain: the domain is all real numbers.

Key Concepts

Addition of Functions

When working with the addition of functions, we are essentially combining two functions into one by adding their corresponding outputs. Consider two functions, say \( f(x) \) and \( g(x) \). To find the sum of these functions, which is expressed as \((f+g)(x)\), you simply add the outputs of \( f(x) \) and \( g(x) \) for each input \( x \).

For instance, if \( f(x) = 5x + 2 \) and \( g(x) = 3x + 4 \), then the addition of these functions is done as follows:

• Add the expressions: \( (f+g)(x) = f(x) + g(x) = (5x + 2) + (3x + 4) \)
• Combine like terms to simplify: \( (f+g)(x) = (5x + 3x) + (2 + 4) = 8x + 6 \)

As a result, the function \((f+g)(x)\) is \( 8x + 6 \). By mastering addition of functions, you build a foundation for more complex algebraic operations.

Subtraction of Functions

Subtraction of functions works similarly to addition, but instead we subtract the value of one function from the other. To find \((f-g)(x)\) for the functions \( f \) and \( g \), you subtract the output of \( g(x) \) from the output of \( f(x) \) for each input \( x \).

Using the same function examples, \( f(x) = 5x + 2 \) and \( g(x) = 3x + 4 \), here is how subtraction is performed:

• Set up the subtraction: \( (f-g)(x) = f(x) - g(x) = (5x+2) - (3x+4) \)
• Distribute the negative sign and combine like terms: \( (f-g)(x) = 5x - 3x + 2 - 4 = 2x - 2 \)

In this case, the resulting function \((f-g)(x)\) is \( 2x - 2 \).
Understanding how to correctly perform subtraction with functions is essential for solving many types of algebraic problems.

Domain of Functions

The domain of a function refers to all the possible input values \( x \) that will result in a valid output for the function. For the functions \( f(x) = 5x + 2 \) and \( g(x) = 3x + 4 \), both are linear functions, which means they can accept any real number as an input without any restrictions.

This property carries over to their sum and difference. Hence, both \((f+g)(x)\) and \((f-g)(x)\) are also defined for all real numbers. In this context, there are no limitations or restrictions in the domain.

To summarize, for both the addition and subtraction of linear functions, the domain typically includes all real numbers. Understanding the domain is key to knowing where the function behaves correctly and where it might encounter issues.