Homework Help: Determine if these are functions

1. Nov 18, 2012

nicnicman

Hello everyone,
I just want to make sure I'm doing these problems correctly. Here they are

Are the following functions?

1. F : Z→Z where F(x) = 4/7x + 1
Answer: Not a function. F(1) is not an integer.

2. G : R→R where G(x) = {2x + 2 if x ≥ 0, x - 3 if x ≤ 0
Answer: Not a function, because x is not well-defined. For G(0) there are two outputs.

3. h : R→R where h(x) = { x^3 if x > 3, 2x - 3 if x ≤ 3

Any suggestions are welcome.

Thanks.

2. Nov 18, 2012

micromass

That's correct.

3. Nov 18, 2012

nicnicman

Thanks for the help!

4. Nov 18, 2012

HallsofIvy

Just a nitpick- in 1 F certainly IS a function, just not from Z to Z.

5. Nov 18, 2012

nicnicman

Good point. Maybe I should specify. Would something like this work?
For Z→Z, F(x) = 4/7x + 1 is not a function.

6. Nov 18, 2012

haruspex

For once, I disagree with Halls. F:Z→Z was part of the definition you were given, so the definition as a whole is not a valid definition of a function.

7. Nov 18, 2012

micromass

I agree with this. The domain and the codomain are an essential part of a function.

8. Nov 18, 2012

Ray Vickson

In 1: be careful, use parentheses. It makes a difference whether you mean F(x) = (4/7)x + 1 or F(x) = 4/(7x) + 1. In the first case F(x) is an integer whenever x is an integer multiple of 7, but in the second case F(x) is never an integer for any nonzero integer value of x.

RGV

9. Nov 20, 2012

nicnicman

Ray, since the domain and codomain are defined as Z→ Z wouldn't either case, F(x) = (4/7)x + 1 or F(x) = 4/(7x) + 1, still have to produce an integer for every integer x?

Also, the function should have been written as F : Z→ Z where F(x) = (4/7)x + 1.

Last edited: Nov 20, 2012