The intuitive idea of a function

A function must satisfy two main requirements for its outputs: existence and uniqueness. Existence means that for every input in the stated domain, the function must produce an output.

Conditions: The domain is explicitly stated.; The function rule is well-defined for all inputs in the domain.

The squaring function $f(x)=x^2$ maps different inputs to the same output if those inputs are opposites (e.g., $2$ and $-2$). This does not violate the function rule because a function requires exactly one output for each input, but it allows multiple inputs to share the same output.

Conditions: The function is $f(x)=x^2$.; Inputs are distinct real numbers.

The vertical line test is a graphical method used to determine whether a curve represents a function of $x$. A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Conditions: The graph is plotted on a standard Cartesian coordinate system.; The independent variable is $x$.

For the squaring function $f(x)=x^2$ defined on all real numbers, the domain is the set of all real numbers ($\mathbb{R}$), meaning every input is allowed. The range is the set of all nonnegative real numbers ($[0, \infty)$), because squaring any real number results in a value greater than or equal to zero.

Conditions: The function is $f(x)=x^2$.; The domain is specified as all real inputs.