Problem 15 - Algebra 2

Question

Let \(f(x)=2 x^{2}+x-3\) and \(g(x)=x-1 .\) Perform each function operation and then find the domain. $$ f(x)-g(x) $$

Step-by-Step Solution

Answer

The result of the function operation \(f(x)-g(x)\) is \(2x^{2} - 2\), and its domain is all real numbers.

Step 1: Perform the function operation

The operation to be performed is \(f(x)-g(x)\). Subtract \(g(x)\) from \(f(x)\) to perform the operation. So,\[f(x)-g(x) = \left(2x^{2} + x - 3\right) - (x - 1).\] First, distribute the negative sign throughout the \(g(x)\) function. Then, consolidate like terms.

Step 2: Simplify the result

The equation becomes:\[f(x)-g(x) = 2x^{2} + x - 3 - x + 1.\]Simplify the equation further:\[f(x)-g(x) = 2x^{2} + 0x - 2.\] Thus, \[f(x)-g(x) = 2x^{2} - 2.\]

Step 3: Find the Domain

The domain of a function is the set of all possible input values (x-values), which will produce a valid output. A polynomial function, such as these functions quotient, has a domain of all real numbers, because you can substitute any real number for x and the operation will still be valid. Consequently, the domain of \(f(x)-g(x)\) is all real numbers.

Key Concepts

Function Operations

When working with polynomial functions like \( f(x) = 2x^2 + x - 3 \) and \( g(x) = x - 1 \), performing function operations is a fundamental skill.Let's explore this concept by considering the operation \( f(x) - g(x) \). Here, you subtract one polynomial from another. This process involves distributing the subtraction through each term in the second polynomial.For instance, our original algebraic expression: \[(2x^2 + x - 3) - (x - 1)\]requires you to apply the negative sign to each term within \( g(x) \). This gives: \[(2x^2 + x - 3) - x + 1\]From here, you consolidate like terms, ensuring only similar, like terms combine: - The \( x \)'s cancel each other out, resulting in 0.- Constants \( -3 + 1 = -2 \).This leaves us with the simplified expression,\[ 2x^2 - 2\].Always remember, operations involving functions require attention to detail, particularly with the distribution of operations like subtraction.

Domain of Functions

The domain of a function is crucial because it describes all the possible values that can be plugged into the function without encountering any computational errors, such as dividing by zero or taking the square root of a negative number (in the real number system).For polynomial functions like the ones in our example,\[f(x) - g(x) = 2x^2 - 2\],it’s straightforward. Polynomial functions do not have restrictions such as division by zero, which might limit the domain of other types of functions.Here, the domain of \( f(x) - g(x) \) is all real numbers:

• Polynomials are defined for every input value of \( x \).
• No crucial limitations such as square roots or variables in the denominator.
• This leads to an unrestricted domain.

So you can confidently input any real number, making the domain of \( f(x) - g(x) \) all real numbers.

Simplifying Expressions

Simplifying expressions is about making them more manageable while maintaining their original meaning.The goal in our example:\[ (2x^2 + x - 3) - (x - 1)\] was to combine terms and reduce complexity.

Steps to Simplify:

- **Distribute:** Carefully apply operations, like the subtraction sign, across all relevant terms.- **Combine Like Terms:** Like terms are those that have the same variable raised to the same power. In our case, \( x - x \) simplifies to zero, and constants \(-3 + 1\) combine to \(-2\).- **Rewrite:** Simplify the expression by collecting results to reach \[2x^2 - 2\].Always check your work step-by-step to ensure accuracy. Simplification is often required and it's a fundamental skill for ensuring clarity in mathematics.