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mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Equation As Piecewise Function?

Consider the equation below which has an upper and lower section.

Upper section: y = 1 if x is rational.

Bottom section: y = 0 if x is irrational.

A. Is this a function?
B. If so, what is its domain and range?
C. Find the x and y intercepts, if any.
D. If it is a function, is it even, odd or neither?
E. How would you describe its graph?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,637

Re: Equation As Piecewise Function?

This is tough to determine.  Both the rationals and the irrationals occur everywhere along the number line. I think it is possible to construct a rational that lies between two irrationals, and an irrational between pairs of rationals (I need to check this*).

So the y values keep switching between 0 and 1. (domain and range : easy )

There is a y axis intercept. x axis intercepts keep occurring.

Is it well defined .... if so it is a function.  It won't be an odd function as it has no  negative y values. Is it even? You need to do the usual test.

If you were able to graph it, it would appear as two horizontal lines but this is an illusion as it is discontinuous everywhere.

Bob

* I'll come back with a more definite answer when I've had time to check.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Equation As Piecewise Function?

This is tough to determine.  Both the rationals and the irrationals occur everywhere along the number line. I think it is possible to construct a rational that lies between two irrationals, and an irrational between pairs of rationals (I need to check this*).

So the y values keep switching between 0 and 1. (domain and range : easy )

There is a y axis intercept. x axis intercepts keep occurring.

Is it well defined .... if so it is a function.  It won't be an odd function as it has no  negative y values. Is it even? You need to do the usual test.

If you were able to graph it, it would appear as two horizontal lines but this is an illusion as it is discontinuous everywhere.

Bob

* I'll come back with a more definite answer when I've had time to check.

I will let you work on that one, Bob. In the textbook, the equation is set up like a piecewise function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,637

Re: Equation As Piecewise Function?

I can prove the four results: between two rationals there is always a rational and an irrational, and between two irrationals is always a rational and an irrational.

Not sure if you want that much but it's been good for me.

Can you do any parts now?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Equation As Piecewise Function?

I can prove the four results: between two rationals there is always a rational and an irrational, and between two irrationals is always a rational and an irrational.

Not sure if you want that much but it's been good for me.

Can you do any parts now?

Bob

Let me try. Be back later with the right or wrong answers.

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Equation As Piecewise Function?

I can prove the four results: between two rationals there is always a rational and an irrational, and between two irrationals is always a rational and an irrational.

Not sure if you want that much but it's been good for me.

Can you do any parts now?

Bob

Visit the following link below. Someone answered this question at algebra.com but some of what he or she said is not clear to me. What do you make of the answer given?

See here:

https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1139600.html

Bob

Administrator

Registered: 2010-06-20

Posts: 10,637

Re: Equation As Piecewise Function?

I agree with all parts.  Post which bits are unclear.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Equation As Piecewise Function?

I agree with all parts.  Post which bits are unclear.

Bob

Sometimes, all it takes is a little searching online.