We have the following functions:
[tex]\begin{gathered} g(x)=\sqrt[\square]{x-4} \\ h(x)=2x-8 \end{gathered}[/tex]
Step 1. Calculate the composition of the functions:
[tex]g\circ h[/tex]
which is defined as follows:
[tex]g\circ h=g(h(x))[/tex]
Thus, we need to substitute h(x) into the x in g(x):
[tex]g\circ h=\sqrt[]{(2x-8)-4}[/tex]
Step 2. Simplify the expression:
[tex]\begin{gathered} g\circ h=\sqrt[]{2x-8-4} \\ g\circ h=\sqrt[]{2x-12} \end{gathered}[/tex]
Step 3. Calculate the restrictions on the domain.
The domain of a function are the possible values for the variable x.
In this case, since we have a square root, we can only have possitive values inside of the square root.
Thus, we need 2x-12 to be equal or greater to 0:
[tex]2x-12\ge0[/tex]
Step 4. Solve the inequality for x:
[tex]\begin{gathered} 2x-12\ge0 \\ 2x\ge12 \\ x\ge\frac{12}{2} \\ x\ge6 \end{gathered}[/tex]
Answer:
x≥6
