Let's find the [tex]\( x \)[/tex]-intercept of the given function [tex]\( h(x) = 2 \sqrt[3]{x - 10} + 4 \)[/tex].
The [tex]\( x \)[/tex]-intercept of a function occurs where the value of the function is zero. That is, we need to solve for [tex]\( x \)[/tex] when [tex]\( h(x) = 0 \)[/tex].
Starting with the equation:
[tex]\[ h(x) = 2 \sqrt[3]{x - 10} + 4 \][/tex]
we set it equal to zero:
[tex]\[ 2 \sqrt[3]{x - 10} + 4 = 0 \][/tex]
First, isolate the term involving the cube root on one side of the equation:
[tex]\[ 2 \sqrt[3]{x - 10} = -4 \][/tex]
Next, divide both sides by 2 to simplify:
[tex]\[ \sqrt[3]{x - 10} = -2 \][/tex]
To solve for [tex]\( x \)[/tex], we need to get rid of the cube root by cubing both sides of the equation:
[tex]\[ (\sqrt[3]{x - 10})^3 = (-2)^3 \][/tex]
This simplifies to:
[tex]\[ x - 10 = -8 \][/tex]
Finally, solve for [tex]\( x \)[/tex] by adding 10 to both sides:
[tex]\[ x = -8 + 10 \][/tex]
[tex]\[ x = 2 \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercept of the function [tex]\( h(x) = 2 \sqrt[3]{x - 10} + 4 \)[/tex] is [tex]\( x = 2 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
