Question

two functions G and F on defined in the figure below. find the domain and range of the composition of f o g.write your answers in set notation. Edit: please if possible double-check the answers just to be safe

Answer 1

In set notation:

- Domain of
`\( f \circ g \): \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \)`

- Range of
`\( f \circ g \): \( \{0, 1, 2, 3, 5\} \)`

In the composition of two functions,
`\( f \circ g \)`, we first apply
`\( g \)` and then
`\( f \)` to the result. The domain of the composition will be the set of all inputs for
`\( g \)` that map to outputs of
`\( g \)` which are in the domain of
`\( f \)`. The range of the composition will be the set of all possible outputs from
`\( f \)` after
`\( g \)`has been applied.

To find the domain of
`\( f \circ g \)`, we look at the domain of
`\( g \)` and see where those values map to in
`\( f \)`. The domain of
`\( g \)` is the set of all x-values in its left oval, and the range is the set of all values it maps to in the right oval, which then must be in the domain of
`\( f \)`.

From the figure provided:

- The domain of
`\( g \)` appears to be {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

- The range of
`\( g \)` is {0, 1, 2, 3, 4, 5, 6} since these are the values
`\( g \)` maps to.

- The domain of
`\( f \)` is {0, 1, 2, 3, 4, 5, 6}.

Since the range of
`\( g \)` matches the domain of
`\( f \)`, the domain of
`\( f \circ g \)` will be the same as the domain of
`\( g \)` because all outputs from
`\( g \)` are valid inputs for
`\( f \).`

Therefore, the domain of
`\( f \circ g \) is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.`

To find the range of
`\( f \circ g \)`, we look at the range of
`\( f \)` since it is the last function applied. However, we need to consider only the values of the range of
`\( g \)` that are also in the domain of
`\( f \).`

- The range of
`\( f \)` appears to be {0, 1, 2, 3, 5} since these are the values
`\( f \)` maps to from its domain.

However, we have to map the values from the domain of
`\( g \)` through
`\( f \)` to get the range of
`\( f \circ g \).`

Mapping the domain of
`\( g \)` through
`\( f \)`, we get the range of
`\( f \circ g \)` as the values that
`\( f \)` maps to, which appear to be {0, 1, 2, 3, 5}. Hence, the range of
`\( f \circ g \)` is {0, 1, 2, 3, 5}.

In set notation:

- Domain of
`\( f \circ g \): \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \)`

- Range of
`\( f \circ g \): \( \{0, 1, 2, 3, 5\} \)`

Answer 2

Step 1 :

In this question, we are given that two functions G and F on defined in the figure below.

We are meant to find the domain and range of the composition of f o g.

Step 2 :

Comparing the domain of f with the range of g means that

we are comparing the numbers that are common for both,

as we can see 0 and 6 are common for both.

Step 3 :

`\begin{gathered} \text{The domain of f o g is given as :}\lbrace\text{ 4, 5, 7 }\rbrace \\ \text{The range of f o g is given as : }\lbrace1\text{ }\rbrace \end{gathered}`

CONCLUSION:

`\begin{gathered} \text{The domain of f o g is given as :}\lbrace\text{ 4, 5, 7 }\rbrace \\ \text{The range of f o g is given as :}\lbrace1\text{ }\rbrace \end{gathered}`