Question

State the domain restriction(s) in interval notation of \displaystyle f\left(g\left(x\right)\right)f(g(x)) given: \displaystyle f\left(x\right)=\sqrt{3x-2}f(x)= 3x−2 ​ and \displaystyle g\left(x\right)=x-7g(x)=x−7

Answer 1

The interval notation for the domain is
`[(23)/(3),\infty ]`.

Explanation:

Consider the provided information.

It is given that
`\:f\left(x\right)=√(3x-2),\:\text{ and }\:g\left(x\right)=x-7`

We need to find the value of
`f\left(g\left(x\right)\right)`.

Put the value of g(x) in
`f\left(g\left(x\right)\right)`.

`f\left(g\left(x\right)\right)=f(x-7)` ....(1)

Now, put x=x-7 in
`\:f\left(x\right)=√(3x-2)`

`\:f\left(x-7\right)= √(3(x-7)-2)`

`\:f\left(x-7\right)= √(3x-21-2)`

`\:f\left(x-7\right)= √(3x-23)`

From equation 1.

`f\left(g\left(x\right)\right)=\:f\left(x-7\right)= √(3x-23)`

The domain of the function is the set of input values for which a function is defined.

Here, the value of
`3x-23` should be greater or equal to 0 as the square root of a negative number is not real.

Domain=
`3x-23\geq0`

`x\geq(23)/(3)`

The value of x is all real number greater than
`(23)/(3)`.

Hence, the interval notation for the domain is
`[(23)/(3),\infty ]`.