Questions: Consider the following functions. f(x) = sqrt(x-4) and g(x) = sqrt[3]x Find the formula for (f+g)(x) and simplify your answer. Then find the domain for (f+g)(x). Round your answer to two decimal places, if necessary.

Solution Steps

To find the formula for \((f+g)(x)\), we need to add the two given functions: \(f(x) = \sqrt{x-4}\) and \(g(x) = \sqrt[3]{x}\). The domain of \((f+g)(x)\) is determined by the intersection of the domains of \(f(x)\) and \(g(x)\). For \(f(x)\), \(x\) must be greater than or equal to 4, and for \(g(x)\), \(x\) can be any real number. Therefore, the domain of \((f+g)(x)\) is \(x \geq 4\).

We have the functions defined as follows: \[ f(x) = \sqrt{x - 4} \] \[ g(x) = \sqrt[3]{x} \] To find \((f+g)(x)\), we add the two functions: \[ (f+g)(x) = f(x) + g(x) = \sqrt{x - 4} + \sqrt[3]{x} \] Thus, the formula for \((f+g)(x)\) is: \[ (f+g)(x) = \sqrt{x - 4} + x^{1/3} \]

The expression \((f+g)(x)\) does not simplify further in a meaningful way, so we retain it as: \[ (f+g)(x) = x^{1/3} + \sqrt{x - 4} \]

The domain of \(f(x)\) requires that: \[ x - 4 \geq 0 \implies x \geq 4 \] The function \(g(x)\) is defined for all real numbers. Therefore, the domain of \((f+g)(x)\) is determined by the more restrictive condition from \(f(x)\): \[ \text{Domain of } (f+g)(x) = [4, \infty) \]

Final Answer