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

Answer:

The interval notation for the domain is $[\frac{23}{3},\infty ]$.

Step-by-step explanation:

Consider the provided information.

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

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 $\:f\left(x\right)=\sqrt{3x-2}$

$\:f\left(x-7\right)= \sqrt{3(x-7)-2}$

$\:f\left(x-7\right)= \sqrt{3x-21-2}$

$\:f\left(x-7\right)= \sqrt{3x-23}$

From equation 1.

$f\left(g\left(x\right)\right)=\:f\left(x-7\right)= \sqrt{3x-23}$

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= $3x-23\geq0$

$x\geq\frac{23}{3}$

The value of x is all real number greater than $\frac{23}{3}$.

Hence, the interval notation for the domain is $[\frac{23}{3},\infty ]$.