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. Step 1: Understand the operation (ƒ-g)(x), which means subtracting the function g(x) from the function ƒ(x).
2. Step 2: Write the expression for (ƒ-g)(x) as ƒ(x) - g(x) = \sqrt{4x-1} - \frac{1}{x}.
3. Step 3: Determine the domain of ƒ(x) = \sqrt{4x-1}. The expression under the square root, 4x-1, must be greater than or equal to zero. Solve the inequality 4x-1 \geq 0 to find the domain of ƒ(x).
4. Step 4: Determine the domain of g(x) = \frac{1}{x}. The denominator x cannot be zero, so solve the inequality x \neq 0 to find the domain of g(x).
5. Step 5: Find the intersection of the domains of ƒ(x) and g(x) to determine the domain of (ƒ-g)(x). This involves combining the conditions from Step 3 and Step 4.

Key Concepts

Function Operations

Function operations involve combining two functions to create a new function. In this case, (ƒ-g)(x) represents the difference between the functions ƒ(x) and g(x). To find this, you subtract the output of g(x) from the output of ƒ(x) for any given x. Understanding how to perform these operations is essential for solving the problem.

Domain of a 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 a specific domain. For g(x)=1/x, the function is undefined when x=0, which also restricts its domain. Identifying the domain is crucial for ensuring valid inputs in function operations.

Square Root Function

The square root function, denoted as √(x), returns the non-negative value whose square equals x. In the context of ƒ(x)=√(4x-1), it is important to recognize that the expression inside the square root must be greater than or equal to zero. This affects both the domain and the behavior of the function, as it cannot produce negative outputs, which is a key consideration when performing operations with other functions.