Understanding Functions: One-to-One and Onto Mappings

This video tutorial explains how to classify mappings as one-to-one or onto functions. It begins with a review of function definitions, then delves into the characteristics of one-to-one (injective) and onto (surjective) functions. The tutorial provides examples of mappings, analyzing each to determine if they are functions, and if so, whether they are one-to-one, onto, both, or neither. The video concludes with additional examples to reinforce the concepts discussed.

What is a key characteristic of a one-to-one function?

• Different inputs have the same output.

• Every output has at most one input.

• Every input has exactly one output.

• Every input has multiple outputs.

Which of the following is true for an onto function?

• Every input has exactly one output.

• Some elements in the output set have no corresponding inputs.

• Every input has multiple outputs.

• Every element in the output set has at least one corresponding input.

Why is a mapping with an input having two outputs not considered a function?

• It violates the rule of having exactly one output per input.

• It is both one-to-one and onto.

• It has more outputs than inputs.

• It has more inputs than outputs.

In a one-to-one function, what is true about the elements in the output set?

• Each element in the output set has multiple corresponding inputs.

• Each element in the output set has no corresponding inputs.

• Each element in the output set has at most one corresponding input.

• Each element in the output set has exactly two corresponding inputs.

What condition must be met for a function to be both one-to-one and onto?

• Every input has multiple outputs.

• Every output has multiple inputs.

• Every output has exactly one corresponding input.

• Every input has no corresponding output.

If a mapping has an element in the output set with zero corresponding inputs, what can be concluded?

• The mapping is one-to-one.

• The mapping is onto.

• The mapping is neither one-to-one nor onto.

• The mapping is both one-to-one and onto.

What is a characteristic of a mapping that is one-to-one but not onto?

• Every input has no corresponding output.

• Every input has multiple outputs.

• Some elements in the output set have no corresponding inputs.

• Every element in the output set has at least one corresponding input.

In the context of functions, what does 'injective' mean?

• The function is onto.

• The function is one-to-one.

• The function is neither one-to-one nor onto.

• The function is both one-to-one and onto.

What is the result when a mapping has an element in the output set with two corresponding inputs?

• The mapping is both one-to-one and onto.

• The mapping is onto.

• The mapping is one-to-one.

• The mapping is neither one-to-one nor onto.

What can be concluded if a mapping is neither one-to-one nor onto?

• It has elements in the input set with multiple outputs.

• It meets the criteria for both one-to-one and onto functions.

• It is a valid function.

• It has elements in the output set with no corresponding inputs.