Example below:

Two functions f and g are defined in the figure below. 2 3 3 5 7 7 7 Domain of f Range of f Range of g Find the domain and range of the composition g of. Write your answers in set notation. (a) Domain of g of : (b) Range of g of : EXPLANATION Given functions f and g, we define the composition of g with f, written g of, as follows. ( 8 of ) (x) = 8(f(xx)) So, to evaluate g of at an input x, we first find f (x). More Then, we use f (x) to find g (f (x)). (a) For the current problem, a sketch of the composition of g with f is shown in the figure below.V Range of f Espan Range of g g Domain of f Aa Domain of g Note that the range of f is not equal to the domain of g. For instance, 2 is an element of the domain of f, but f (2) = 8 is not an element of the domain of g. So, the value g (f (2)) is not defined. We can do a similar analysis with the other elements of the domain of f. f (3) =3 is an element of the domain of g f (5) =0 is an element of the domain of g f (7) =3 is an element of the domain of g f (9) =7 is an element of the domain of g Therefore, the domain of g of is (3, 5, 7, 9}. (b) To find the range of g of, we find the images of those values under g of, as follows. (g of ) (3) = 8(f (3))= 8(3) =4 (g of ) (5) = 8(f (5))= 8(0) =8 (g of ) (7) = 8(f (7)) =8(3) =4 (g of) (9) = 8(f(9)) = 8(7) =4Therefore, the range of g of is (4, 8}. The function g of can be summarized as follows. U 7 9 Domain of g of Range of g of ANSWER (a) Domain of g of :