For the pair of functions defined, find (ƒ+g)(x). Give the domain of each. See Example 2. ƒ(x)=√(4x-1), g(x)=1/x

Function addition involves combining two functions by adding their outputs for each input value. For the functions ƒ(x) and g(x), the sum (ƒ+g)(x) is defined as ƒ(x) + g(x). This operation requires that both functions are defined for the same input values, which is crucial for determining the overall domain of the resulting function.

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For ƒ(x)=√(4x-1), the expression under the square root must be non-negative, leading to the condition 4x-1 ≥ 0. For g(x)=1/x, the function is undefined when x=0. The overall domain of (ƒ+g)(x) is the intersection of the domains of ƒ(x) and g(x).

The square root function, denoted as √(x), returns the non-negative value whose square equals x. It is only defined for non-negative inputs, meaning that any expression under the square root must be greater than or equal to zero. In this case, the function ƒ(x)=√(4x-1) requires that 4x-1 ≥ 0, which directly influences the domain of the function.