Consider the function f:R toR then which of the following is true? A) f is one one if the graph of f intersects some line parallel to X axis in at least two points. B)f is not surjective if the graph f does not intersects atleast one line parallel to X axis C)f is surjective if the graph of f intersects some lines parallel to X axis?

Question

Answer

Introduction:
We are given a function f from the set of real numbers to the set of real numbers, i.e., f: R to R. We need to analyze the properties of the function based on its graph intersecting lines parallel to the x-axis.

A) f is one-one if the graph of f intersects some line parallel to the x-axis in at least two points:
To determine if the function is one-one, we need to check if every element in the domain has a unique image in the co-domain. If the graph of f intersects some line parallel to the x-axis in at least two points, it means that there are two distinct points on the graph with the same y-coordinate. This violates the one-one property, as two distinct elements in the domain map to the same element in the co-domain. Therefore, this statement is false.

B) f is not surjective if the graph of f does not intersect at least one line parallel to the x-axis:
To determine if the function is surjective, we need to check if every element in the co-domain has a pre-image in the domain. If the graph of f does not intersect at least one line parallel to the x-axis, it means that there is at least one y-value in the co-domain that does not have a corresponding x-value in the domain. This violates the surjective property, as there exists an element in the co-domain that is not mapped from any element in the domain. Therefore, this statement is true.

C) f is surjective if the graph of f intersects some lines parallel to the x-axis:
To determine if the function is surjective, we need to check if every element in the co-domain has a pre-image in the domain. If the graph of f intersects some lines parallel to the x-axis, it means that for every y-value in the co-domain, there exists at least one x-value in the domain that maps to it. This satisfies the surjective property, as every element in the co-domain has a corresponding element in the domain. Therefore, this statement is true.

Conclusion:
Based on the analysis, the correct statements are:
- Statement B: f is not surjective if the graph of f does not intersect at least one line parallel to the x-axis.
- Statement C: f is surjective if the graph of f intersects some lines parallel to the x-axis.

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