Question

Find the domain and the range of the function. $$f(x)=\sqrt{x+3}$$

Short answer

The domain of the function \(f(x)=\sqrt{x+3}\) is \([-3, \infty)\) and the range is \([0, \infty)\).

Step-by-step solution

Identify Domain

The function is \(f(x)=\sqrt{x+3}\). The square root of a number is not defined for negative values. Therefore, the value of \(x\) must be such that \(x+3\) is non-negative or \(x+3 \geq 0\). Solving this inequality, we find that \(x \geq -3\). Hence, the domain is all real numbers \(x\) such that \(x \geq -3\). In interval notation, this is \([-3, \infty)\).

Identify Range

The square root function \(f(x)= \sqrt{x+3}\) always returns a non-negative number, if \(x\) is non-negative. Therefore, for all \(x \geq -3\), \(f(x)\) will produce a non-negative output value. This implies that the function \(f(x)\) can take all the non-negative real numbers as its output. Hence, the range is all non-negative real numbers, which is \([0, \infty)\) in interval notation.