How to check a graph if it is one-to-one function?

Question

Is it by drawing a vertical line?

What id you draw a horizontal line?

Answer 1

No vertical line should intersect the graph twice.
Because then a single point x is projected on multiple points y and you don't have a one-to-one relation.

If you draw a horizontal line you can check if the inverse function is also a one-to-one function.
If this is true than a function is called a bijective function (if it is false than it is a injective function).

Answer 2

Start at the left end of the graph and draw an imaginary vertical line. If it intersects the graph at just ONE point, slide the vertical line to the right while observing whether it still intersects the graph at just one point. If it intersects at just one point for each horizontal position of the vertical line, the graph represents a one-to-one function. This is easy to do by eye alone, no math required.

A horizontal line will not work.

Answer 3

You have a function where y is a function of x

Imagine a vertical line (parallel to the y axis) that "scans" the graph, wherever the function is defined.
The function is one-to-one if, at any value of x, the vertical line is only crossed once by the graph. (In other words, for any value of x, there is only one possible value for y).

The square root is not one-to-one, unless the function was defined as being the absolute value. The square root of 4 can be +2 or -2. The vertical line, when placed at x=4, would be crossed at both points (+2 and -2) by the graph.
Using only the absolute value (i.e., only the positive value) makes the square root a one-to-one function

Imagine a horizontal line that can "scan" the graph by moving up and down. If, for any given value of y in the , the line is crossed once, and only once, by the graph, then the function is said to be "onto", meaning that for any function value, there can only be one initial value for x.

The square root, even though it is not one-to-one (unless defined as absolute value), is still onto, for any non-negative value of y.
y = SQRT(x)
If y = 2, then the only possible value for x is +4.
Move the horizontal line to y=-2; then the only possible value for x is +4.


Vertical line: checks if function is one-to-one
Horizontal line: checks if function is onto.

Answer 4

If the graph is one-one, if you draw a vertical line and it does not intercect more than one point with the graph.

When, you plot a graph, on x-axis is the value of variable and on y-axis its the value of function. So, if the line intersect more than one point, it means the function has more than same value for more than one point on x-axis.

I hope that it is clear now.

All the best.

Answer 5

If you can draw a vertical line that intersects the graph two or more times, then it is not one to one, because one x value produces two y values.