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 subtract the function g(x) from ƒ(x). So, write the expression as (ƒ - g)(x) = ƒ(x) - g(x).
2. Substitute the given functions into the expression: (ƒ - g)(x) = \sqrt{4x - 1} - \frac{1}{x}.
3. Next, find the domain of each function separately. For ƒ(x) = \sqrt{4x - 1}, the expression inside the square root must be greater than or equal to zero, so set up the inequality 4x - 1 \geq 0 and solve for x.
4. For g(x) = \frac{1}{x}, the denominator cannot be zero, so x \neq 0. This restriction must be considered when determining the domain of (ƒ - g)(x).
5. Finally, combine the domain restrictions from both functions to find the domain of (ƒ - g)(x). This means taking the intersection of the domain of ƒ(x) and the domain of g(x), excluding any values that make either function undefined.

Key Concepts

Function Operations (Addition and Subtraction)

Function operations involve combining two functions using addition, subtraction, multiplication, or division. For (ƒ - g)(x), subtract the output of g(x) from ƒ(x) for each x in the domain. This creates a new function whose value depends on both original functions.

Domain of a Function

The domain is the set of all input values (x) for which a function is defined. When combining functions, the domain of the resulting function is the intersection of the individual domains, considering restrictions like square roots requiring non-negative radicands and denominators not equal to zero.

Square Root and Rational Function Restrictions

For ƒ(x) = √(4x - 1), the expression inside the square root must be ≥ 0, 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.