For the pair of functions defined, find (ƒ+g)(x), (ƒ-g)(x), (ƒg)(x), and (f/g)(x).Give the domain of each. ƒ(x)=4x²+2x, g(x)=x²-3x+2

Verified step by step guidance

1. First, write down the given functions: \(f(x) = 4x^2 + 2x\) and \(g(x) = x^2 - 3x + 2\).
2. To find \((f+g)(x)\), add the two functions: \((f+g)(x) = f(x) + g(x) = (4x^2 + 2x) + (x^2 - 3x + 2)\). Combine like terms to simplify.
3. To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\): \((f-g)(x) = f(x) - g(x) = (4x^2 + 2x) - (x^2 - 3x + 2)\). Be careful to distribute the minus sign and then combine like terms.
4. To find the product \((fg)(x)\), multiply the two functions: \((fg)(x) = f(x) imes g(x) = (4x^2 + 2x)(x^2 - 3x + 2)\). Use the distributive property (FOIL) to expand the product.
5. To find the quotient \((f/g)(x)\), divide \(f(x)\) by \(g(x)\): \((f/g)(x) = \frac{f(x)}{g(x)} = \frac{4x^2 + 2x}{x^2 - 3x + 2}\). The domain excludes values where \(g(x) = 0\), so solve \(x^2 - 3x + 2 = 0\) to find these excluded values.

Key Concepts

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division. For example, (ƒ+g)(x) means adding the outputs of ƒ(x) and g(x) for each x. Understanding how to perform these operations is essential to manipulate and analyze combined functions.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When combining functions, the domain is restricted to values common to both functions and, in the case of division, excludes values that make the denominator zero.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents with coefficients. Both ƒ(x) and g(x) are polynomials, which are defined for all real numbers, simplifying domain considerations except when division is involved.