MHB Why Restrict the Domain for Certain Functions?

[mathdad]

Why must we restrict the domain of certain functions to solve the problem?

 

[skeeter]

Why must we restrict the domain of certain functions to solve the problem?

solve what problem?

 

[mathdad]

For example, find the range of y = sqrt{x + 4}.

 

[MarkFL]

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?

 

[mathdad]

I am still not too clear, Mark. The question should be what is a ONE-TO-ONE FUNCTION?

 

[MarkFL]

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.

 

[mathdad]

The textbook has an entire section on different types of functions.