That is simply a result of the definition of a function. A function is a mapping such that for each value of x there is only one value of y.
Test it by the vertical line test. That is, if a vertical line passes through the two points of the graph, this graph is not the graph of a function.
points
The function is not continuous.
The function y = x is the graph that passes from the points (-1, -1), (0, 0), and (1, 1) The function y = 4x is the graph that passes form the points (-1, -4), (0, 0), and (1, 4) Sketch these graphs in a same x and y coordinate system, and you can see both of them
We set the denominator to zero to find the singularities: points where the graph is undefined.
You find the average rate of change of the function. That gives you the derivative on different points of the graph.
Test it by the vertical line test. That is, if a vertical line passes through the two points of the graph, this graph is not the graph of a function.
points
A graph of a function cannot have two different coordinates (or points) with the same x-value because, by definition, a function assigns exactly one output (y-value) for each input (x-value). If a graph did have two points with the same x-coordinate but different y-coordinates, it would violate the definition of a function, as a single input would yield multiple outputs. This concept is often referred to as the "vertical line test," where any vertical line drawn on the graph intersects it at most once.
The vertical line test can be used to determine if a graph is a function. If two points in a graph are connected with the help of a vertical line, it is not a function. If it cannot be connected, it is a function.
Yes it is possible. The solutions for a quadratic equation are the points where the function's graph touch the x-axis. There could be 2 places to that even if the graph looks different.
If a vertical line, within the domain of the function, intersects the graph in more than one points, it is not a function.
If a vertical line, within the domain of the function, intersects the graph in more than one points, it is not a function.
The "vertical line test" will tell you if it is a function or not. The graph is not a function if it is possible to draw a vertical line through two points.
Since there are no "following" points, none of them.
The function is not continuous.
They mean the graph/function is decreasing.