Let's determine the values of the function [tex]\( h(x) \)[/tex] at [tex]\( x = 0 \)[/tex] and [tex]\( x = 4 \)[/tex] by analyzing the definition of [tex]\( h \)[/tex].
1. For [tex]\( x = 0 \)[/tex]:
According to the piecewise function definition, when [tex]\( 0 \leq x < 4 \)[/tex], the function is given by [tex]\( h(x) = 2x^2 - 3x + 10 \)[/tex].
Therefore, to find [tex]\( h(0) \)[/tex]:
[tex]\[
h(0) = 2(0)^2 - 3(0) + 10 = 10
\][/tex]
So, [tex]\( h(0) = 10 \)[/tex].
2. For [tex]\( x = 4 \)[/tex]:
According to the piecewise function, when [tex]\( x \geq 4 \)[/tex], the function is given by [tex]\( h(x) = 2^x \)[/tex].
Therefore, to find [tex]\( h(4) \)[/tex]:
[tex]\[
h(4) = 2^4 = 16
\][/tex]
So, [tex]\( h(4) = 16 \)[/tex].
Thus, the values of the function are:
[tex]\[
\begin{array}{l}
h(0) = 10 \\
h(4) = 16
\end{array}
\][/tex]
