Why Restrict the Domain for Certain Functions?

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Discussion Overview

The discussion revolves around the necessity of restricting the domain of certain functions, particularly in the context of finding inverses of functions that are not one-to-one. Participants explore the implications of domain restrictions on function behavior and the concept of one-to-one functions.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the specific problem that requires domain restriction.
  • Another participant provides an example of the function y = sqrt{x + 4} to illustrate the concept of range and domain restriction.
  • A participant explains that restricting the domain of a function that is not one-to-one allows for the determination of its inverse, using the function f(x) = x^2 as an example.
  • Some participants express confusion regarding the definition and characteristics of one-to-one functions.
  • A later reply defines a one-to-one function as one where each output corresponds to a unique input, noting that a graph crossing a horizontal line at more than one point indicates it is not one-to-one.
  • One participant mentions that the textbook contains a section on different types of functions, suggesting a resource for further exploration.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the concept of one-to-one functions and the necessity of domain restrictions. There is no consensus on the clarity of these concepts, indicating ongoing confusion and exploration.

Contextual Notes

Some assumptions about the definitions of functions and their properties remain unexamined, and the discussion does not resolve the mathematical implications of domain restrictions fully.

mathdad
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Why must we restrict the domain of certain functions to solve the problem?
 
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RTCNTC said:
Why must we restrict the domain of certain functions to solve the problem?

solve what problem?
 
For example, find the range of y = sqrt{x + 4}.
 
I assume you're talking about restricting the domain of a function that is not one-to-one to a domain on which the function is one-to-one in order to find the inverse of that function. A function takes an input, and maps it to an output. The function's inverse is a mapping of the original function's outputs to its inputs.

Consider the function:

$$f(x)=x^2$$

It's not one-to one because it maps both $a$ and $-a$ to $a^2$. So, without restricting the domain, the inverse would map $a^2$ to two inputs (the outputs for the inverse), and a function should map it a unique output for any given input. So, if we restrict the domain of $f$ to $0\le x$ or $x\le0$, there will be no ambiguity in the outputs of the inverse function.

Does that make sense?
 
I am still not too clear, Mark. The question should be what is a ONE-TO-ONE FUNCTION?
 
RTCNTC said:
I am still not too clear, Mark. The question should be what is a ONE-TO-ONE FUNCTION?

A one-to-one function is a function that has only one input mapping to any particular output. If the graph of a function crosses any horizontal line at more than one point, then it is not one-to-one. It will have no "turning points," or places where it goes from increasing to decreasing or from decreasing to increasing.
 
The textbook has an entire section on different types of functions.
 

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