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

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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).

Substitute the given functions into the expression: (ƒ - g)(x) = \(\sqrt{4x - 1}\) - \(\frac{1}{x}\).

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.

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).

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.

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주요 개념

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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.

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