Question

F(x)= 4x-1 and g(x) = sqr root -x+6(f◦g)(x)=give domain=(g◦f)(x)= give domain

Answer 1

The given functions are

`\begin{gathered} f(x)\text{ = }4x\text{ - 1} \\ g(x)\text{ = }\sqrt[]{-\text{ x + 6}} \end{gathered}`

To find (fog)(x), we would substitute x = g(x) into f(x). We have

`(fog)(x)\text{ = 4(}\sqrt[]{-\text{ x + 6}})\text{ - 1}`

To find the domain of (fog)(x), we would write the value inside the square root as an inequality that is greater than or equal to zero and solve for x. We have

- x + 6 ≥ 0

- x ≥ - 6

Dividing both sides of the inequality by - 1, we have

x ≤ 6

We would exclude any value greater than 6 from the domain. Thus,

Domain = (- infinity, 6]

To find (gof)x), we would substitute x = f(x) = 4x - 1 into g(x). We have

`\begin{gathered} (\text{gof)(x) = }\sqrt[]{-\text{ (4x - 1) }+\text{ 6}}\text{ = }\sqrt[]{-\text{ 4x + 1 + 6}}\text{ } \\ (\text{gof)(x) = }\sqrt[]{-\text{ 4x + 7}} \end{gathered}`

To find the domain, we have

- 4x + 7 ≥ 0

- 4x ≥ - 7

Dividing both sides of the inequality by - 4, we have

x ≤ 7/4

We would exclude any value greater than 7/4 from the domain. Thus,