Problem 15 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The function is \(g-f = -x - 4\) with domain \(\mathbb{R}\).

Step 1: Understand the Problem

We need to find the function that represents \(g-f\), which means we need to subtract function \(f(x)\) from \(g(x)\). We also need to determine the domain of this new function.

Step 2: Substitute Functions

Substitute the given functions into the expression for \(g-f\). The expression becomes:\[g(x) - f(x) = (x-3) - (2x+1)\]

Step 3: Simplify the Expression

Simplify the expression for \(g-f\):\[(x-3) - (2x+1) = x - 3 - 2x - 1 = -x - 4\]Thus, the function \(g-f\) is \(-x - 4\).

Step 4: Determine the Domain

Both \(f(x) = 2x + 1\) and \(g(x) = x - 3\) are linear functions. Linear functions are defined for all real numbers. Therefore, the subtraction \(-x - 4\) is also defined for all real numbers. Thus, the domain of \(g-f\) is all real numbers, \(\mathbb{R}\).

Key Concepts

Subtracting Functions

When dealing with functions in algebra, sometimes you have to perform operations such as addition, subtraction, multiplication, or division. Subtracting functions is a common operation and is essentially a straightforward process.
To subtract one function from another, say function \(f(x)\) from \(g(x)\), you simply take all of \(f(x)\) and subtract it from all of \(g(x)\). You need to pay attention to the signs while subtracting every term.
In our exercise, \(g(x) = x - 3\) and \(f(x) = 2x + 1\), the subtraction \(g-f\) involves subtracting each component of \(f(x)\) from the corresponding parts of \(g(x)\). This gives the expression:

• \(g(x) - f(x) = (x - 3) - (2x + 1)\)
• Make sure to distribute the negative sign appropriately: \((x - 3) - 2x - 1\)
• Finally, simplify: \(-x - 4\).

The result is a new function \(-x - 4\) which represents \(g(x) - f(x)\). Remember, when subtracting functions, it's crucial to distribute and manage signs carefully to achieve an accurate result.

Simplifying Expressions

Simplifying an expression means rewriting it in a simpler or more efficient form. This often involves combining like terms and following the order of operations.
For the subtraction of our functions, \( (x-3) - (2x+1) \), simplifying requires a few key steps:

• Remove parentheses. Just remember—be careful with signs when you distribute the subtraction: \(x - 3 - 2x - 1\).
• Next, combine like terms. Group all the \(x\)'s together and all constant terms together: \(x - 2x\) and \(-3 - 1\).
• This yields \(-x - 4\), which is the simplified version of our expression.

By carefully organizing the terms and performing correct arithmetic, you can streamline expressions swiftly and accurately.

Domain of Functions

The domain of a function encompasses all the input values (\(x\) values) that the function can handle without encountering issues. For instance, you may encounter fractions where the denominator cannot be zero, or square roots must not take negative numbers.
However, both \(f(x) = 2x + 1\) and \(g(x) = x - 3\) are linear functions. A linear function is defined for every real number, meaning there are no restrictions like those seen in fractions or square roots.
Consequently, the subtraction \(g-f = -x - 4\) inherits this all-encompassing domain from its linear components.

• The domain for \(g-f\) is \(\mathbb{R}\), the set of all real numbers.
• This means the function never runs into any undefined areas, ensuring it can take on any real number as an input value without trouble.

Always consider the original form of functions in operations to determine whether there are any restrictions to their domain.