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Question Video: Identifying Injective Functions Mathematics

True or False: If 𝑓 and 𝑔 are both one-to-one functions, then 𝑓 + 𝑔 must be a one-to-one function.

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Video Transcript

True or false: If 𝑓 and 𝑔 are both one-to-one functions, then 𝑓 plus 𝑔 must be a one-to-one function.

In order to prove that this statement is true, we must prove it is true for all possible functions in this case. However, in order to prove a statement is false, we can do so via a counterexample. We begin by recalling that a function is one-to-one if each element of the range of the function corresponds to exactly one element of the domain. In mapping diagrams, this means that each element of the range has exactly one arrow pointing to it.

Let’s consider the functions 𝑓 and 𝑔 represented by the mapping diagrams shown. We see that both functions 𝑓 and 𝑔 are injective or one-to-one because each element of the range has exactly one arrow from the domain pointed at it. When we add 𝑓 and 𝑔 as required, we obtain the following diagram. Noting that all three lines in the range give the same number, we can redraw the mapping diagram. Since the range element 10 has three arrows pointing to it, this range element corresponds to more than one of the domain elements. We can therefore conclude the function 𝑓 plus 𝑔 is not injective. Using a counterexample, we have shown that the statement β€œif 𝑓 and 𝑔 are both one-to-one functions, then 𝑓 plus 𝑔 must be a one-to-one function” is false.

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