Find Domain and Range of Composite Function

Question

Given

$$f(x) = \{(-5,0),(-4,-2),(-2,3),(1,5),(4,2)\}$$

$$g(x) = \{(0,-2),(8,4),(-2,5),(5,-5),(3,1)\}$$

$1$) Find the domain and range of $f(g(x))$.

$2$) Find the domain and range of $g(f(x)$.

Answer 1

First, let me address your notation. You should have:

$$f = \{(-5,0),(-4,-2),(-2,3),(1,5),(4,2)\}$$

$$g = \{(0,-2),(8,4),(-2,5),(5,-5),(3,1)\}$$

Then for example, the fact that $(-5,0)\in f$ is the same as saying $f(-5)=0.$

Now, there are a few ways to go about this problem. Since $f$ and $g$ are "small" functions (in the sense of cardinality), it's probably simplest to find the compositions directly. Namely, $$f\circ g:=\{(x,z)\mid \exists y,(x,y)\in g,(y,z)\in f\}$$ and $$g\circ f:=\{(x,z)\mid \exists y,(x,y)\in f,(y,z)\in g\}.$$ for instance, $(0,3)\in f\circ g,$ since $(0,-2)\in g$ and $(-2,3)\in f.$ For both compositions, the domain will be the set of first components; the range, of second components.

Answer 2

Good points:

$$x\in\text{dom}(f\circ g)\to\exists z\big((x,z)\in f\circ g\big)\to \exists z,~\exists y \left((x,y)\in g\wedge(y,z)\in f\right)$$

$$z\in\text{ran}(f\circ g)\to\exists x\left((x,z)\in f\circ g\right)\to \exists x,~\exists y \left((x,y)\in g\wedge(y,z)\in f\right)$$