Respuesta :
Answer:
The interval notation for the domain is [tex][\frac{23}{3},\infty ][/tex].
Step-by-step explanation:
Consider the provided information.
It is given that [tex]\:f\left(x\right)=\sqrt{3x-2},\:\text{ and }\:g\left(x\right)=x-7[/tex]
We need to find the value of [tex]f\left(g\left(x\right)\right)[/tex].
Put the value of g(x) in [tex]f\left(g\left(x\right)\right)[/tex].
[tex]f\left(g\left(x\right)\right)=f(x-7)[/tex] ....(1)
Now, put x=x-7 in [tex]\:f\left(x\right)=\sqrt{3x-2}[/tex]
[tex]\:f\left(x-7\right)= \sqrt{3(x-7)-2}[/tex]
[tex]\:f\left(x-7\right)= \sqrt{3x-21-2}[/tex]
[tex]\:f\left(x-7\right)= \sqrt{3x-23}[/tex]
From equation 1.
[tex]f\left(g\left(x\right)\right)=\:f\left(x-7\right)= \sqrt{3x-23}[/tex]
The domain of the function is the set of input values for which a function is defined.
Here, the value of [tex]3x-23[/tex] should be greater or equal to 0 as the square root of a negative number is not real.
Domain= [tex]3x-23\geq0[/tex]
[tex]x\geq\frac{23}{3}[/tex]
The value of x is all real number greater than [tex]\frac{23}{3}[/tex].
Hence, the interval notation for the domain is [tex][\frac{23}{3},\infty ][/tex].
