? Combining Functions Find f + g, f - g, fg, and f/g and their domains. f(x)=x^2+2 x, g(x)=3 x

step 1 We're going to calculate several combinations of the functions. F of X is equal to x squared plu

? Combining Functions Find f + g, f - g, fg, and f/g and...

Key Concepts

Function Operations

Function operations refer to the methods by which two or more functions are combined using operations such as addition, subtraction, multiplication, and division. These operations involve applying the algebraic operations directly to the outputs of the functions for a given input, resulting in a new function that represents the desired combined behavior.

Addition and Subtraction of Functions

Adding or subtracting functions involves combining the corresponding outputs. For any two functions f and g, (f + g)(x) is found by f(x) + g(x) and (f - g)(x) is found by f(x) - g(x). This process is performed on the expressions defining f and g, leading to a new algebraic expression that represents the sum or difference, respectively.

Multiplication of Functions

Multiplying functions means creating a new function that is the product of the outputs of the individual functions, hence (f * g)(x) = f(x) * g(x). This operation combines the algebraic expressions of f and g by multiplication, often resulting in a more complex expression that requires careful handling of algebraic terms.

Division of Functions

Dividing one function by another, represented as (f/g)(x), involves taking the quotient f(x)/g(x). The primary consideration in this operation is ensuring that the denominator is never zero, which imposes restrictions on the domain of the resulting function. This aspect is critical because division by zero is undefined.

Domain Considerations

The domain of a combined function is determined by the domains of the individual functions and any additional restrictions introduced by the operations. For addition, subtraction, and multiplication, the domain is typically the intersection of the domains of the individual functions. However, for division, it is essential to also exclude any input values that make the denominator equal to zero.

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