Question

f(x)= -3x -6 and g(x) = √-x+4 Determine each of the following.(f°g)(x)=Give the domain of (f°g)(x)(g°f)(x)=Give the domain (g°f)(x)

Answer 1

PART A

We are given the following values f(x)= -3x -6 and g(x) = √-x+4

The function (fog)(x) would be gotten by inserting g(x) for the value of x in f(x)

`\begin{gathered} (\text{fog)(x)}=f(g(x)) \\ =-3(\sqrt[]{-x+4})-6 \\ =-3(\sqrt[]{4-x})-6 \end{gathered}`
`\text{fog(x)}=3(\sqrt[]{4-x})-6`

We can find the domain of (f°g)(x) by first finding the domain of g(x)

The domain of g(x) is the values of x for which g(x) is defined.

g(x) is defined for the values of x below

`x\le4`

We then also need to find the domain of fog

fog(x) will not exist for the values of x below:

`x\le4`

Since this coincides with the domain of g(x), therefore the domain of fog(x) is

`x\le4`

PART B

The function (gof)(x) would be gotten by inserting f(x) for the value of x in g(x)

`\begin{gathered} \text{gof(x)}=g(f(x) \\ =\sqrt[]{4-(-3x-6)} \\ =\sqrt[]{3x+10} \\ \end{gathered}`

Therefore, the value of gof(x)

`gof(x)=\sqrt[]{3x+10}`

We can find the domain of (g°f)(x) by first finding the domain of f(x)

The domain of f(x) is the values of x for which f(x) is defined.

f(x) is defined for the values of x below

`-\infty\le x\le\infty`

The above indicates f(x) is defined for all real numbers

We then also need to find the domain of gof

gof(x) will not exist for the values of x below:

`x<-(10)/(3)`

Excluding the above from the domain of f(x)

We would therefore have the domain of gof(x) as

`x\ge-(10)/(3)`