# Functions - Range vs Codomain

Gold Member
Hi. This isnt directly a homework question, but it will help in general.

Im having a little trouble understanding what the difference between a range and codomain is. For example, for the function $$f(x) = \frac{3}{{2x - 2}}$$, i understand that the domain is $$\{ x \in R:x \ne 1\}$$. Now, i also believe that the possible values that can be outputted by the function is given by
$$\{ f(x) \in R:f(x) \ne 0\}$$. Is this the codomain or range?

Dan.

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dextercioby
Homework Helper
The range/codomain is the image of the domain through the function

$$\mbox{Ran}(f(x)):=\left\{ f(x)\left|\right x\in D(f(x)) \right\}$$

In your case, first make a plot of the function first.

Hi. This isnt directly a homework question, but it will help in general.

Im having a little trouble understanding what the difference between a range and codomain is. For example, for the function $$f(x) = \frac{3}{{2x - 2}}$$, i understand that the domain is $$\{ x \in R:x \ne 1\}$$. Now, i also believe that the possible values that can be outputted by the function is given by
$$\{ f(x) \in R:f(x) \ne 0\}$$. Is this the codomain or range?

Dan.
that is the range of the function. codomain is usually a superset (sometimes equal as well) of the range. its generally defined in the question itself, like f:R-->R (here both domain and codomain are the set of real nos.),but range will be a subset(or an equal set) of R depending upon the function definition.

HallsofIvy
Homework Helper
According to Wikipedia, the "codomain" of a function f:X-> Y is the set Y. The "range" is the subset of Y that f actually maps something onto.

For example, if f:R->R is defined by f(x)= ex, then the "codomain" is R but the "range" is the set, R+, of all positive real numbers.

Notice that you cannot tell the "codomain" of a function just from its "formula". I could just as easily define f:R->R+, with f(x)= ex. Now the codomain and domain would be the same.

Notice that you cannot tell the "codomain" of a function just from its "formula". I could just as easily define f:R->R+, with f(x)= ex. Now the codomain and domain would be the same.
I have a doubt, I think we also cannot tell what the "domain" is just from the "formula" . We can say what it "is not" but we cant say what it "is".

for instance we can define a function as f:[1,2]->R , with f(x) = ex . here "domain" is what we define(i.e [1,2]) ,"co-domain" is what we define(i.e R) , but "range" is obtained from the formula, which in this case would be [e,e^2]

but the formula definitely can tell us what domain is not.
ex :- f(x) = $$\sqrt{x}$$ we can't say domain is R. we have to define domain as R$$^{+}$$ or it's subsets.