For the pair of functions defined, find (f/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, find the expression for the quotient of the two functions, which is given by \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). Substitute the given functions to get \(\left(\frac{f}{g}\right)(x) = \frac{\sqrt{4x - 1}}{\frac{1}{x}}\).
2. Simplify the expression by dividing by a fraction, which is equivalent to multiplying by its reciprocal. So, \(\left(\frac{f}{g}\right)(x) = \sqrt{4x - 1} \times x\).
3. Next, determine the domain of \(f(x) = \sqrt{4x - 1}\). Since the expression inside the square root must be non-negative, set up the inequality \$4x - 1 \geq 0\( and solve for \)x$.
4. Determine the domain of \(g(x) = \frac{1}{x}\). Since division by zero is undefined, exclude \(x = 0\) from the domain.
5. Finally, find the domain of \(\left(\frac{f}{g}\right)(x)\) by combining the restrictions from both \(f(x)\) and \(g(x)\). This means the domain includes all \(x\) values that satisfy \$4x - 1 \geq 0\( and \)x \neq 0$.

Key Concepts

Function Division (f/g)(x)

Dividing two functions f and g means creating a new function (f/g)(x) = f(x) / g(x). This requires evaluating both functions at the same input x and then dividing the results, provided g(x) ≠ 0 to avoid division by zero.

Domain of a Function

The domain of a function is the set of all input values x for which the function is defined. For composite functions like (f/g)(x), the domain is the intersection of the domains of f and g, excluding values that make the denominator zero or cause undefined expressions.

Square Root Function Domain Restrictions

For a function involving a square root, such as f(x) = √(4x - 1), the expression inside the root must be non-negative (≥ 0) to produce real outputs. This restriction limits the domain to values of x satisfying 4x - 1 ≥ 0.