Find ƒ+g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

Verified step by step guidance

1. Identify the given functions: \(f(x) = \sqrt{x + 4}\) and \(g(x) = \sqrt{x - 1}\).
2. Find the sum of the functions by adding them: \((f + g)(x) = f(x) + g(x) = \sqrt{x + 4} + \sqrt{x - 1}\).
3. Determine the domain of \(f(x)\) by setting the expression inside the square root to be greater than or equal to zero: \(x + 4 \geq 0\) which simplifies to \(x \geq -4\).
4. Determine the domain of \(g(x)\) similarly: \(x - 1 \geq 0\) which simplifies to \(x \geq 1\).
5. Find the domain of \((f + g)(x)\) by taking the intersection of the domains of \(f(x)\) and \(g(x)\), which is all \(x\) values satisfying both \(x \geq -4\) and \(x \geq 1\). This means the domain is \(x \geq 1\).

Key Concepts

Function Addition (ƒ + g)

Adding two functions involves creating a new function where each output is the sum of the outputs of the original functions at the same input. For functions f and g, (ƒ + g)(x) = f(x) + g(x). This operation combines the values of both functions pointwise.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the domains of the original functions, ensuring all expressions are valid.

Square Root Function Domain Restrictions

A square root function √(expression) is defined only when the expression inside the root is non-negative. For f(x) = √(x + 4), x + 4 ≥ 0; for g(x) = √(x − 1), x − 1 ≥ 0. These inequalities determine the domain restrictions for each function.