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A function is a mathematical relationship between two sets of elements, usually represented as x and y, in which each element from the x set (domain) is associated with exactly one element from the y set (range). The domain of a function is the set of all possible input values (x-values) for which the function is defined that can be assigned to the independent variable.

The formula of a function is an equation that describes the relationship between the input values (x) and the output values (y). This equation shows how to calculate the output for a given input.

Let's consider two functions, f(x) and g(x), both defined by the same formula, say \(y = x^2\), but with different domains. Domain of f(x) = {x | -3 ≤ x ≤ 3} Domain of g(x) = {x | 0 ≤ x ≤ 4}

Now that we have defined the two functions, we will graph them using their respective formula and domain. To do this, we will list the corresponding x and y values within their domains and plot them on a coordinate plane. f(x): x = -3, y = 9 x = -2, y = 4 x = -1, y = 1 x = 0, y = 0 x = 1, y = 1 x = 2, y = 4 x = 3, y = 9 g(x): x = 0, y = 0 x = 1, y = 1 x = 2, y = 4 x = 3, y = 9 x = 4, y = 16

When we plot these points on the coordinate plane and then connect the dots to form a smooth curve, we will find that the two graphs are different in appearance. The function f(x) will appear as a parabola with a vertex at the origin (0, 0) and extending from -3 to 3 on the x-axis. The function g(x), on the other hand, will also appear as a parabola but only in the first quadrant (where both x and y values are positive) with its initial point at the origin (0, 0) and final point at (4,16). In conclusion, even though the two functions have the same formula, their graphs differ because of their different domains. The domain restricts the portion of the graph that is shown, and as a result, the two functions have different appearances when plotted on the coordinate plane.