91Ó°ÊÓ

The square root function is represented mathematically by the notation \(\sqrt{x}\). This function only produces non-negative outputs when provided with a non-negative input. In simpler terms, the square root of any number will always be zero or positive.

When we add a constant to the square root function, as in \(f(x) = \sqrt{x} + 3\), we shift the entire graph of the function upwards by that constant amount. The domain of \(\sqrt{x}\) remains the same, covering only non-negative values. This shift is why the function never becomes zero, hence it's impossible for \(\sqrt{x} + 3\) to equal zero. Solving for this would suggest \(\sqrt{x} = -3\), which is not possible because the square root of a number cannot produce a negative result.

Complex numbers extend the idea of real numbers to include imaginary numbers. They are usually written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).

Despite the useful and extensive application of complex numbers in different areas of mathematics, the condition \(\sqrt{x} = -3\) cannot hold true for any real or complex number \(x\). This is because the square root function outputs only non-negative values; there is no real or imaginary number that will satisfy \(\sqrt{c} = -3\). Therefore, complex numbers do not provide a solution to our function \(f(x) = \sqrt{x} + 3\) equating to zero.

The domain of a function represents all the possible inputs (values of \(x\)) for which the function is defined. For the function \(f(x) = \sqrt{x} + 3\), the domain is restricted to non-negative real numbers \(x \geq 0\) because the square root of a negative number is not defined within the set of real numbers.

The range of a function represents all possible outputs (values of \(f(x)\)). For \(f(x) = \sqrt{x} + 3\), since \(\sqrt{x}\) can range from 0 to positive infinity, adding 3 shifts this range to begin from 3 to positive infinity. Thus, the range of the function is \ [3, \infty)\. This explains why \(f(x)\) can never be zero or negative, reinforcing the fact that there is no real or complex number \(c\) such that \(f(c) = 0\).

The constraints established by the domain and range reaffirm the understanding that the function \(f(x) = \sqrt{x} + 3\) can never satisfy the condition hinted at in the original problem.