Domain & Range of f: Identifying Math Function

[needhelp83]

Identify the domain and range for the function

f= {(x,y)\in \mathbb{R} x \mathbb{R}:y=\chi _{\mathbb{Z}}(x)}


f={(x,y)\in \mathbb{R} x \mathbb{R}:y=\frac{e^x + e^{-x}}{2}


How do I determine the correct results. I am not really understanding the terminology.

 

[HallsofIvy]

For the first one, I surely don't! Is it really "chi" and "z" of x? \chi _{\mathbb{Z}}? If those are not defined in your text I can't help you.

For the second, that is also known as the "hyperbolic cosine", cosh(x), but you don't really need to know that. Are you familiar with the function e^x? You should know that its domain (values of x for which it is defined) is all real numbers while its range (possible values of the function itself) is all positive numbers. It should be easy to get the domain and range of (e^x+ e^-x)/2 from that.

 

[needhelp83]

From my textbook the \chi_{A} (x) is called the characteristic function of A.

 

[Dick]

From my textbook the \chi_{A} (x) is called the characteristic function of A.

Then if Z means the integers, and f(x)=y, then f(x)=1 if x is an integer and f(x)=0 if x is not an integer. So what are the domain and range of that?

 

[needhelp83]

Alright this should be right

1. Domain = \mathbbc{R}
Range = {1}
2. Domain = \mathbbc{R}
Range = [1,x)

 

[needhelp83]

Is this done correctly?

 

[HallsofIvy]

Alright this should be right

1. Domain = \mathbbc{R}
Range = {1}

No, as Dick said, this function can have values of either 0 or 1. The range is {0, 1}.

2. Domain = \mathbbc{R}
Range = [1,x)

The range is a set of numbers. It cannot involve the variable x. The range is [1, \infty). Was that a typo?