Problem 82 - Elementary Algebra

Question

Calculate \((f+g)(x)\) and \((f-g)(x)\) and state the restrictions to the domain. \(f(x)=1 x-1\) and \(g(x)=1 x+5\)

Step-by-Step Solution

Answer

\((f+g)(x) = 2x + 4\), \((f-g)(x) = -6\), domain is all real numbers.

Step 1: Understanding the Functions

The functions given are \( f(x) = 1x - 1 \) and \( g(x) = 1x + 5 \). These are linear functions. We will now work on the sum and difference of these functions.

Step 2: Calculating \((f+g)(x)\)

The function \((f+g)(x)\) is found by adding the functions \(f(x)\) and \(g(x)\). Thus, \((f+g)(x) = (1x - 1) + (1x + 5)\). Combining like terms, we get \((f+g)(x) = 2x + 4\).

Step 3: Calculating \((f-g)(x)\)

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). Therefore, \((f-g)(x) = (1x - 1) - (1x + 5)\). Simplifying, \((f-g)(x) = 1x - 1x - 1 - 5\), which results in \((f-g)(x) = -6\).

Step 4: Domain Considerations

Both functions \(f(x)\) and \(g(x)\) are linear, meaning they are defined for all real numbers. There are no restrictions such as division by zero or square roots of negative numbers involved. Hence, the domain for both \((f+g)(x)\) and \((f-g)(x)\) is all real numbers, \(x \in \mathbb{R}\).

Key Concepts

Function Addition

When we talk about function addition, we mean combining two functions to form a new function by adding their outputs. It is comparable to how numbers are added together to get a sum. For two linear functions, such as the ones we are dealing with:

• \(f(x) = 1x - 1\)
• \(g(x) = 1x + 5\)

The sum \((f+g)(x)\) is found by simply adding these functions:

• \((f+g)(x)=(1x - 1) + (1x + 5)\)
• Combine like terms to simplify it: \(2x + 4\)

Function addition results in another function, which also appears as a straight line, given the original functions are linear.
Thus, the new function is also a linear function with a slope of 2 and a y-intercept of 4.

Function Subtraction

Function subtraction follows the same principle as function addition, but instead of adding, we are subtracting the outputs of two functions. When given linear functions \(f(x)\) and \(g(x)\), you find their difference \((f-g)(x)\) by subtracting one function from the other:

• \((f-g)(x) = (1x - 1) - (1x + 5)\)
• Simplify by removing like terms: \(0x - 6\) or simply \(-6\)

Interestingly, the result here is a constant function, meaning the output is the same no matter what \(x\) value is used.
This specific subtraction resulted in a horizontal line at \(y = -6\).

Domain of Functions

Both function addition and subtraction involve considering the domain of the resulting new function. The domain of a function is the complete set of all possible input values (\(x\) values) that will allow the function to work without error. For linear functions like these, there are generally no restrictions, as they do not include operations that might limit the domain such as division by zero or taking square roots of negative numbers.
Therefore:

• Both functions \(f(x) = 1x - 1\) and \(g(x) = 1x + 5\) are defined for all real numbers.
• Similarly, the domains of their sum and their difference, \((f+g)(x) = 2x + 4\) and \((f-g)(x) = -6\), are also all real numbers: \(x \in \mathbb{R}\).

These domains are straightforward because linear equations continue in both directions on the x-axis infinitely.