Why Restrict the Domain for Certain Functions?

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SUMMARY

This discussion centers on the necessity of restricting the domain of certain functions to accurately determine their inverses. Specifically, the function f(x) = x² is highlighted as a non-one-to-one function, which requires domain restriction to either 0 ≤ x or x ≤ 0 to ensure a unique output for each input. The concept of one-to-one functions is defined, emphasizing that such functions do not cross horizontal lines at multiple points, thus maintaining a unique mapping from inputs to outputs.

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  • Familiarity with the concept of one-to-one functions
  • Basic knowledge of inverse functions
  • Graphical interpretation of functions and their behaviors
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  • Study the properties of one-to-one functions in detail
  • Learn about inverse functions and their calculations
  • Explore domain restrictions in various types of functions
  • Review graphical methods for identifying function behaviors
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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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