Problem 13 - 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 function (f + g)(x) = 2x^2 + 2x - 4 and its domain is all real numbers.

Step 1: Perform Function Operation

To add two functions, we simply add each component of the given functions. So, for the given functions f(x) = 2x^2 + x - 3, and g(x) = x - 1, their sum (f + g)(x) becomes: (f + g)(x) = f(x) + g(x) = (2x^2 + x - 3) + (x - 1) = 2x^2 + 2x - 4.

Step 2: Determine the Domain of the Resulting Function

For (f + g)(x) = 2x^2 + 2x - 4, which is a polynomial, the domain is all real numbers. This is because there is no restriction in the function which requires the exclusion of any real number values from the domain. Hence, the domain of (f + g)(x) is all real numbers.

Key Concepts

Function Operations

Function operations involve performing mathematical processes, such as addition, subtraction, multiplication, and division, on two or more functions. In this exercise, we focus on adding two functions.

• To add functions, you simply add like terms. For the functions given:
• \(f(x) = 2x^2 + x - 3\)
• \(g(x) = x - 1\)
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The sum is found by combining these terms:

• \((f + g)(x) = f(x) + g(x) = (2x^2 + x - 3) + (x - 1)\)

This results in a new function:

• \((f + g)(x) = 2x^2 + 2x - 4\)

This process maintains the structure of a polynomial function. Adding functions is straightforward and follows the same rules as adding polynomial expressions.

Domain of Functions

The domain of a function is the set of all possible input values (usually \(x\) values) for which the function is defined. Understanding the domain is crucial, as it tells us the scope of the inputs we can use.

For most polynomial functions, there are no restrictions on the values of \(x\) that you can input, meaning their domain is all real numbers. This is because polynomial functions do not have any operations that would cause undefined values, such as division by zero or taking the square root of a negative number.In this case, the result of our function operation is:

• \((f + g)(x) = 2x^2 + 2x - 4\)

Since this is a simple quadratic polynomial, its domain is all real numbers.Whenever dealing with polynomial functions, it's safe to assume their domain is unrestricted, unless specified otherwise.

Real Numbers

Real numbers encompass all the numbers that can possibly be used in everyday contexts. This set includes several different types of numbers:

• Positive numbers (e.g., 1, 2.5)
• Negative numbers (e.g., -3, -0.75)
• Zero
• Rational numbers (fractions and decimals)
• Irrational numbers (e.g., \(\sqrt{2}\), \(\pi\))

In the context of polynomial functions like \((f + g)(x) = 2x^2 + 2x - 4\), all these real numbers can be used as input values. They form the function's domain.Using real numbers for polynomial functions means that there are no limits to the \(x\) values. This makes them very adaptable and applicable in numerous mathematical scenarios and everyday calculations.