Why is it useful to find the domain and range of a function graph

Question

I know about domain and range but my professor has asked us why it may be useful to find the domain and range and I cant really think of a reason that would be considered "useful". Can anyone think of anything that it can be useful to find domain and range. Maybe just some ideas please of what they may be looking for. Thanks

Answer 1

*Note: $\operatorname{Dom}(f(x))$ denotes the domain of a function while $\operatorname{Ran}(f(x))$ denotes the range of a function.

Domain defines where a specific function $f(x)$ is defined. It is important to find the domain (more important for rational functions) because without domain, one would have to assume that $f(x)$ is defined everywhere. In some situations, that isn't the case. For example, in a rational function like $r(x) = \frac 1x$ ($r: \mathbb{R} \to \mathbb{R}$), we would have to assume it would be also defined for $x = 0$. But, it isn't. $\frac 10$ isn't defined because you can't divide by zero. It is even evident in a graph. If you graph $r(x) = \frac 1x$, there is no y-coordinate for $x = 0$. However, it does get really close. The range helps us see what results you will get from inputting the input that is within a domain. If you input a value that isn't in the domain, it won't have a value within the range. Let's look at another function: $g(x) = \sqrt x + 2$. Let's look at the domain. The domain of the function is $$\operatorname{Dom}(g(x)) = \{x \in \mathbb{R} \mid x \geq 0\}$$ But the range of this is the set of outputs that $g$ returns. The range is $$\operatorname{Ran}(g(x)) = \{g(x) \in \mathbb{R} \mid g(x) \geq 2\}$$

Also, some people tend to get range and codomain confused. The codomain of a function is the set of possible outputs for a function $f(x)$. Howewer, the range (aka the image) is the actual set of outputs you can get from the function. In some cases, the codomain can also be equal to the range; in other cases, it may not.

Bottom line: Not every function is defined everywhere, but to be aware of where it is defined, you should find the domain. To know the outputs, you find the range.