Final Answer:
A function can have a square root and still be a function. The square root function, f(x) = √x, is a valid function because it follows the one input, one output rule. For instance, when f(4) = √4, it yields a single output, either +2 or -2, not both simultaneously.
Explanation:
The definition of a function states that for each input value (x), there can be only one corresponding output value (f(x)). In the case of f(x) = √x, let's consider f(4). The square root of 4 can indeed result in two solutions: +2 and -2. However, for a specific input of 4, the function yields only one value, either +2 or -2, but not both at the same time. This adheres to the principle of a function, where a single input uniquely determines a single output.
Mathematically, √4 = 2 or -2. However, when discussing the function f(x) = √x, f(4) specifically denotes the output for x = 4. Therefore, f(4) = √4 yields a single output value of 2 or -2, not both simultaneously. This distinction emphasizes that while the square root function may produce multiple values for a given number, it maintains the one-to-one correspondence between inputs and outputs required for a function.
The function f(x) = √x exemplifies a function where the square root operation satisfies the function's criteria by associating a unique output for each input, despite the possibility of multiple solutions for certain inputs. Thus, despite the square root having multiple outcomes, it still qualifies as a function due to its adherence to the one-to-one mapping principle between inputs and outputs.