Given the functions f and g, find f+g, f-g, f ·g, and (f)/(g), and state the domain of each.

step 1 We first want to find the sum function f plus g, which really just means f of x plus g of x. So

Given the functions f and g, find f+g, f-g, f ·g, and...

Key Concepts

Arithmetic Operations on Functions

Arithmetic operations on functions involve combining two functions using operations such as addition, subtraction, multiplication, and division. When these operations are performed, the new function is defined by applying the operation to the outputs of the original functions for each input value. This concept is essential for understanding how complex functions can be constructed from simpler ones by combining their effects pointwise.

Determining the Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When performing arithmetic operations on functions, the domain of the resulting function is typically the intersection of the domains of the individual functions. Additionally, for operations like division, it is necessary to exclude any inputs that would result in division by zero, ensuring that the function remains valid and well-defined.

Polynomial Functions

Polynomial functions are expressions consisting of variables and constants combined using addition, subtraction, and multiplication, along with non-negative integer exponents. A key property of polynomial functions is that their domain is all real numbers, as there are no inherent restrictions like division by zero. This concept is foundational in understanding the behavior and properties of many functions encountered in algebra and calculus.

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