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I have searched everywhere on ways to do these Combining Functions problem so here goes. Everywhere I look, this is only explained through actual functions. Ive yet to see one done with only the domain and range of a function.

Let f,g,h be functions with domain and ranges below: f has domain [-1,1) and range [0,2) g has domain [0,2) and range [-1,1) h has domain [1,3) and range [1,2)

For each of the following proposed new functions, specify its domain if it exists, otherwise state that the function does not exist.

(f+g), (f+h), (g o h), (h o g)

Any advice would be a godsend. I already have the answers, just looking for explanations for them.

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  • $\begingroup$ Well, take $f+g$, say. What might that mean? Well, for some $x$ we'd hope to write $(f+g )(x)=f(x)+g(x)$ but for that to make sense we'd need to have both $f(x)$ and $g(x)$ defined. Is that possible? Yes! the intersections in the domains of $f,g$ is $[0,1)$. So $f+g$ has domain $[0,1)$. I don't believe it is possible to specify the range without further information. We know it is contained in $[-1,3)$ but we don't know exactly what it is. the others are similar. $\endgroup$
    – lulu
    Mar 15, 2017 at 11:32
  • $\begingroup$ Should say: the definition of "range" isn't universally agreed on. I believe most people say it means the set of values taken by function. Others say it is simply a set that the function is said to map to, with no assumption that it hits every value. Do you know which definition you are using? $\endgroup$
    – lulu
    Mar 15, 2017 at 11:34
  • $\begingroup$ The Wiki article on Range gives a useful discussion of the ambiguity between the two possible definitions. $\endgroup$
    – lulu
    Mar 15, 2017 at 11:42

1 Answer 1

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(f + g) has domain [0,1) and range [-1,3).

(f + h) has domain [-1,3) and range [1,4).

(g o h) has domain [1,3) and range [-1,1).

(h o g) isn't well defined.(It may or may not exist.)

Note that domain of a function is the set of all values that the function can accept. Asking for the value of a function outside its domain is absurd. The range is the super-set of the outputs for the function.

In the first two cases, for the sum of functions to be defined, I have taken the intersection of the domains of the two functions (for points outside this intersection, at-least one function is going to be not defined). I could have set the domain of the function to $\mathbb R$, but I chose the set such that it is impossible for the function to attain values outside this set.

In the third and fourth problems, for (f o g) to be well defined on the domain of g, the range of g must be a subset of the domain of f. We can also restrict the domain to only include points such that g(x) $\in$ domain(f).

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