Functions, Domains, And equality

[L Huyghe]

Homework Statement

Given an example of two different functions f and g, both of which have the set of real numbers as their domain, such that f(x)=g(x) for every rational number.

2. The attempt at a solution

I have yet to figure a way to approach this problem. Since it appears as though they are only equal for rational numbers.

 

[hgfalling]

It doesn't seem from the problem statement that they are only equal for rational numbers, but in that case, why not just use that in the function definitions? You could define a function that is 0 for all rational numbers and 1 at all irrational numbers, for example. Then come up with a second function that meets the problem criteria.

 

[L Huyghe]

You are right in doesn't necessarily state that all they are not equal in terms of irrational numbers in their domain

 

[HallsofIvy]

Homework Statement

Given an example of two different functions f and g, both of which have the set of real numbers as their domain, such that f(x)=g(x) for every rational number.

2. The attempt at a solution

I have yet to figure a way to approach this problem. Since it appears as though they are only equal for rational numbers.

That is the approach! Define f to be whatever you want, then define g to be that same function on the rationals, but something else on the irrationals. Remember that a "function" does not necessarily mean a single "formula". Such a function cannot be "continuous" anywhere. I do wonder why such a problem would be posted under "precalculus". Since you are the same person who posted the "find f(x) that maps (0, 1) to [0, 1]", what course are these for?

 

[Hurkyl]

The problem does say that f and g are different. So they have to have a different value at one point, at least.

 

[Hurkyl]

I do wonder why such a problem would be posted under "precalculus".

Don't people generally learn the basics of functions in a precalculus class, or earlier?

It sounds like he just did the chapter on piecewise-defined functions. (Tip to opening poster: information such as what lesson you have just learned can be useful both to give you ideas, and to help those who want to help you)

 

[L Huyghe]

Sorry about that, It's probably a good idea to give some more background knowledge about myself. It's not for a course, Just reading precalculus in advance for next year, and I found these problems in the book. I know about piecewise function, but they have yet to be clearly defined in the book yet. These question are in the intro to functions section.

*Also I am still having trouble answering original question, It would be great if u could give me an example. Sorry If i was vague before.

 

[hgfalling]

$$f(x)=\left\{\begin{array}{cc}0,&\mbox{ if } x \in \matbb{Q} \\1, & \mbox{ otherwise }\end{array}\right.$$

$$g(x)=\left\{\begin{array}{cc}0,&\mbox{ if } x \in \matbb{Q} \\14, & \mbox{ otherwise }\end{array}\right.$$