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

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To find (ƒ+g)(x), start by writing the sum of the two functions: (ƒ+g)(x) = ƒ(x) + g(x).

Substitute the given functions into the expression: (ƒ+g)(x) = (2x^2 - 3x) + (x^2 - x + 3).

Combine like terms by adding the coefficients of the same powers of x: (ƒ+g)(x) = (2x^2 + x^2) + (-3x - x) + 3.

Simplify the expression to get the final form of (ƒ+g)(x): (ƒ+g)(x) = 3x^2 - 4x + 3.

Determine the domain of each function. Since both ƒ(x) and g(x) are polynomials, their domains are all real numbers, so the domain of (ƒ+g)(x) is also all real numbers.

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주요 개념

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Function Addition

Function addition involves creating a new function by adding the outputs of two given functions for the same input value x. For functions ƒ(x) and g(x), (ƒ+g)(x) = ƒ(x) + g(x). This operation combines the expressions algebraically to form a single function.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When adding functions, the domain of (ƒ+g)(x) is the intersection of the domains of ƒ(x) and g(x), meaning only values valid for both functions are included.

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 and making addition straightforward.

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