# Domain and Range

1. Jun 29, 2008

### 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.

2. Jun 29, 2008

### HallsofIvy

Staff Emeritus
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 ex? 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 (ex+ e-x)/2 from that.

3. Jun 29, 2008

### needhelp83

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

4. Jun 29, 2008

### Dick

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?

5. Jun 30, 2008

### needhelp83

Alright this should be right

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

6. Jun 30, 2008

### needhelp83

Is this done correctly?

7. Jun 30, 2008

### HallsofIvy

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

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