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Question Video: Identifying One-to-One Functions from Their Graph Mathematics

Is the function shown in the graph a one-to-one function?


Video Transcript

Is the function shown in the graph a one-to-one function?

To begin, we will first recall the general definition of a function and then the specific definition of a one-to-one function. For a relation to be a function, we require each element of the domain to correspond to exactly one element of the range. For that function to be defined as one-to-one, also called an injective function, each element of the range must correspond to exactly one element of the domain. We recall that we can check if a graph is a function by using the vertical line test. This is a visual way of checking that every π‘₯-coordinate of a point on the graph can only have one 𝑦-coordinate. In a Cartesian diagram, the vertical coordinate usually represents the output. Therefore, if a vertical line crosses a graph at more than one point, that means that multiple outputs are assigned to the same input.

For a function to be one-to-one, it must pass the vertical line test and the horizontal line test. Because the horizontal coordinate usually represents the input value, if there is more than one intersection between the horizontal line and the function’s graph, then that range element is associated with more than one domain element. And this would disqualify a function from being one-to-one. Therefore, a function is not one-to-one if there is a horizontal line that crosses its graph more than once.

We are told this graph is a function. But what about the vertical line π‘₯ equals zero? It looks as if the graph may cross that line at multiple places. This can be explained by the existence of a vertical asymptote where the graph approaches but yet is not defined at π‘₯ equals zero. Because this graph is a function, it passes the vertical line test. But we must find out if this graph also passes the horizontal line test.

Over the diagram, we have sketched a horizontal line that crosses the graph at three distinct points. Because we are not given the equation of this function, we could not know the exact coordinates of these three points. However, we can say that the horizontal line that we sketched can be approximately represented by the equation 𝑦 equals 19. And whatever that 𝑦-value is, it intersects three different π‘₯-coordinates. Those π‘₯-coordinates are approximately negative three, zero, and three, and these points all have the same 𝑦-coordinate. We note that the middle point must be slightly to the right of zero, since there are no points defined on the vertical asymptote of π‘₯ equals zero.

In conclusion, this function is not one-to-one because it fails the horizontal line test.

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