For the pair of functions defined, find (ƒ+g)(x), (ƒ-g)(x), (ƒg)(x), and (f/g)(x).Give the domain of each. See Example 2. ƒ(x)=4x^2+2x, g(x)=x^2-3x+2

Verified step by step guidance

1. Step 1: To find \((f+g)(x)\), add the functions \(f(x)\) and \(g(x)\). This means combining like terms from \(f(x) = 4x^2 + 2x\) and \(g(x) = x^2 - 3x + 2\).
2. Step 2: To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). Distribute the negative sign through \(g(x)\) and then combine like terms.
3. Step 3: To find \((fg)(x)\), multiply \(f(x)\) and \(g(x)\) together. Use the distributive property to expand the product \((4x^2 + 2x)(x^2 - 3x + 2)\).
4. Step 4: To find \((f/g)(x)\), divide \(f(x)\) by \(g(x)\). Write it as a fraction \(\frac{4x^2 + 2x}{x^2 - 3x + 2}\) and simplify if possible.
5. Step 5: Determine the domain for each function. For \((f+g)(x)\), \((f-g)(x)\), and \((fg)(x)\), the domain is all real numbers. For \((f/g)(x)\), exclude values that make the denominator zero by solving \(x^2 - 3x + 2 = 0\).

Key Concepts

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, and division. For two functions ƒ(x) and g(x), the operations are defined as (ƒ+g)(x) = ƒ(x) + g(x), (ƒ-g)(x) = ƒ(x) - g(x), (ƒg)(x) = ƒ(x) * g(x), and (f/g)(x) = ƒ(x) / g(x). Understanding these operations is essential for manipulating and analyzing functions.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When performing operations on functions, the domain may change, especially in division, where the denominator cannot be zero. Identifying the domain for each resulting function is crucial to ensure valid outputs.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In this question, both ƒ(x) and g(x) are polynomials. Understanding their structure helps in performing operations and analyzing their behavior, such as finding roots and determining the domain.