Use the given functions f and g to find f+g, f-g, f g, and (f)/(g) . State the domain of each. f

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Use the given functions f and g to find f+g, f-g, f g, and...

Key Concepts

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When performing operations on functions, the domain of the resulting function is generally the intersection of the domains of the component functions, with additional restrictions imposed by the operations (such as excluding inputs that cause division by zero in a quotient). Understanding domain restrictions is essential for accurately describing the set of permissible inputs for any function resulting from arithmetic operations.

Rational Functions and Division by Zero

Rational functions are formed by the ratio of two functions, and their proper consideration involves ensuring that the divisor is not zero. When dividing one function by another, it is crucial to determine the values of the independent variable that would cause the denominator function to be zero, because such values must be excluded from the domain to avoid undefined expressions. This concept teaches the careful handling of potential discontinuities and singularities in function operations.

Arithmetic Operations on Functions

This concept involves performing operations such as addition, subtraction, multiplication, and division on functions, where the operation is applied to the outputs of the functions for each input value. It requires understanding how to combine the expressions and simplify them when possible. The resulting functions are built by performing the arithmetic operations on the original function expressions, term by term, and then determining the appropriate domain based on the operations involved.

Polynomial Functions

Polynomial functions are expressions that consist of sums of powers of the independent variable multiplied by coefficients. They are continuous functions defined for all real numbers unless their domain is otherwise restricted by some operation. Understanding the properties of polynomial functions helps in simplifying arithmetic operations, as these functions are typically smooth and do not have issues like division by zero within their own expressions.

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