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

Verified step by step guidance

1. First, understand that (ƒ+g)(x) means you add the two functions together: (ƒ+g)(x) = ƒ(x) + g(x).
2. Write the sum explicitly using the given functions: (ƒ+g)(x) = \sqrt{4x - 1} + \frac{1}{x}.
3. Next, find the domain of ƒ(x) = \sqrt{4x - 1}. Since the expression inside the square root must be non-negative, set up the inequality: 4x - 1 \geq 0.
4. Solve the inequality for x: 4x \geq 1, so x \geq \frac{1}{4}. This means the domain of ƒ(x) is all real numbers x such that x \geq \frac{1}{4}.
5. Now, find the domain of g(x) = \frac{1}{x}. Since division by zero is undefined, x \neq 0. So the domain of g(x) is all real numbers except x = 0.

Key Concepts

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 and g, (ƒ+g)(x) = ƒ(x) + g(x). This operation combines the two functions pointwise.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the individual domains, considering any restrictions from both functions.

Domain Restrictions from Radicals and Rational Functions

For ƒ(x) = √(4x - 1), the expression inside the square root must be non-negative, so 4x - 1 ≥ 0. For g(x) = 1/x, x cannot be zero because division by zero is undefined. These restrictions determine the valid input values for each function.