Two functions f and g are defined in the figure below. Find...

Two functions $f$ and $g$ are defined in the figure below. Find the domain and range of the composition $g circ f$. Write your answers in set notation. Domain of $g circ f$: Range of $g circ f$:

Transcript

00:01 Okay, we are going to find the domain of f composite g, and we will find the range of f composite g, given we have the function of g and the function of f defined here in the diagrams.

00:22 Okay, what we want to remember is f composite g is equal to f of g of x, where g of x is my input into f.

00:37 So that means x here must be a member of the domain of g, otherwise we cannot find g of x.

00:48 And then g of x needs to be a member of the domain of f, otherwise we cannot find f of x.

00:57 Okay.

00:58 And if you remember, the range of g is really the range of g of g of x, and the range of f is really f of x so knowing that we are going to write out what are all the possible values which is the domain i'm going to call it x of g right here so these are all the possible input values into g of x which is 0 5 6 7 8 9 now if i input into the function g of x, g of x is my output, which is my range.

01:46 When i put zero in, i can obtain four.

01:51 If i input five, i get five out.

01:55 If i put six in, i get zero out.

01:58 If i input seven, i get five.

02:02 If i put in eight, i get seven.

02:05 Nine gives me two.

02:08 Then the next thing we need to look at is what are all the possible x values which are the domain for the function f so now we're going to look at the domain of f so all the possible values there are 0 2, 4, 7, 8, 9.

02:38 Now if i take x and i take 0 and i take 0 and put it in for the x into function g of x i get 4 that means 4 is an input into my function f and there is four is part of the domain of f of x therefore four will be in the domain for f composite g five when i input five in as x i get five out five would become the input into f but there are no fives for my domain therefore that is not within the domain i do the same thing for zero and i notice zero is in the domain of f of x.

03:25 Again, five is not in the domain of f of x.

03:29 7 is in the domain of f of x and two here is in the domain of f of x.

03:36 Okay, so we found out all the possible g of x values that are in the domain of x.

03:42 We can now define our domain for f composite g.

03:50 Those are all the x input values.

03:52 So i have to go back and say, okay, 4 is in the domain of f...

Question

Two functions $f$ and $g$ are defined in the figure below. Find the domain and range of the composition $g circ f$. Write your answers in set notation. Domain of $g circ f$: Range of $g circ f$:

Please give Ace some feedback