Question

Describe the square root function.

Short answer

The square root function \(f(x) = \sqrt{x}\), is only defined for \(x \geq 0\) and gives the square root of \(x\) as output. Its properties include a domain and range of \([0, \infty)\) and a graph that is a rising curve starting at the origin. Common square roots to know include \(\sqrt{1} = 1\), \(\sqrt{4} = 2\), \(\sqrt{9} = 3\), \(\sqrt{16} = 4\).

Step-by-step solution

Definition of Square Root Function

The square root function is a type of radical function, which is defined as \(f(x) = \sqrt{x}\). This function takes an input value, known as \(x\) and gives as output the square root of \(x\). The square root of a number \(x\) is the number that, when squared (multiplied by itself), equals \(x\).

Properties of the Square Root Function

1) The square root function is only defined for \(x \geq 0 \), since the square root of a negative number is not a real number, it is an imaginary number. This means the domain of \(f(x) = \sqrt{x}\) is \([0, \infty)\). \n 2) The output of the square root function is always either positive or zero. This means the range of \(f(x) = \sqrt{x}\) is \([0, \infty)\). \n 3) The graph of the square root function is a rising curve starting at the origin (0,0). It rises more slowly as x increases.

Common Square Root Values

\(\sqrt{1} = 1\), since \(1^2 = 1\). \n \(\sqrt{4} = 2\), since \(2^2 = 4\). \n \(\sqrt{9} = 3\), since \(3^2 = 9\). \n \(\sqrt{16} = 4\), since \(4^2 = 16\). These values should be easily recognized.