Final answer:
To determine if the graph of an equation represents a function of x, we can use the vertical line test. The graph of the equation x = y² is not a function of x because for each x-value, there are two corresponding y-values. The graph of the equation x = y³ is the graph of the function f(x) = x³.
Step-by-step explanation:
The graph of a function is a set of points in the coordinate plane that satisfy a specific relationship between the input and output values. To determine if the graph of an equation represents a function of x, we can use the vertical line test. If a vertical line intersects the graph in more than one point, then the graph does not represent a function of x.
The graph of the equation x = y² is not the graph of a function of x because for each x-value, there are two corresponding y-values (a positive and a negative value). This can be seen by graphing the equation and observing that a vertical line at any x-value intersects the graph in two points.
The graph of the equation x = y³ is the graph of a function of x. This can be confirmed by noting that for each x-value, there is only one corresponding y-value. The equation x = y³ represents the cube function, f(x) = x³. So, the graph of x = y³ is the graph of the function f(x) = x³.
The graph of x = yⁿ, where n is an integer, is the graph of a function of x only when n is an even integer. When n is an odd integer, the graph is not a function of x because for each x-value, there are two corresponding y-values (a positive and a negative value).