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

Verified step by step guidance

1. First, understand that (ƒ - g)(x) means you subtract the function g(x) from ƒ(x). So, write the expression as (ƒ - g)(x) = ƒ(x) - g(x).
2. Substitute the given functions into the expression: (ƒ - g)(x) = (2x^2 - 3x) - (x^2 - x + 3).
3. Distribute the negative sign across the terms in g(x): (ƒ - g)(x) = 2x^2 - 3x - x^2 + x - 3.
4. Combine like terms by grouping the x^2 terms, the x terms, and the constants: (2x^2 - x^2) + (-3x + x) - 3.
5. Determine the domain of each function. Since both ƒ(x) = 2x^2 - 3x and g(x) = x^2 - x + 3 are polynomials, their domains are all real numbers, which can be written as \((-\infty, \infty)\). The domain of (ƒ - g)(x) is also all real numbers.

Key Concepts

Function Operations (Addition and Subtraction)

Function operations involve combining two functions by adding or subtracting their outputs for each input x. For (ƒ - g)(x), subtract g(x) from ƒ(x) to create a new function. This process helps analyze how functions interact and transform.

Domain of a Function

The domain is the set of all input values (x) for which a function is defined. When combining functions, the domain of the resulting function is the intersection of the individual domains. Understanding domain restrictions ensures valid inputs for the combined function.

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. Recognizing polynomial structure aids in function operations.