two functions f and g are defined in the figure below. find the domain and range of the composition g ∘ f…

Answer:

(a) ${1,4,8,9}$ (b) ${6,8}$

Explanation:

Step1: Recall domain - range of composition

The domain of $g\circ f$ is the set of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$. The domain of $f$ is ${1,4,8,9}$. And $f(1) = 3$, $f(4)=4$, $f(8)=6$, $f(9)=7$, all of which are in the domain of $g$. So the domain of $g\circ f$ is ${1,4,8,9}$.

Step2: Find the range of composition

We calculate $g(f(x))$ for each $x$ in the domain of $g\circ f$. $g(f(1))=g(3) = 6$, $g(f(4))=g(4)=6$, $g(f(8))=g(6)$ is not defined, $g(f(9))=g(7)=8$. So the range of $g\circ f$ is the set of all resulting values, which is ${6,8}$.