Let's determine the values of the function [tex]\( h \)[/tex] at the given points [tex]\( x = 0 \)[/tex] and [tex]\( x = 4 \)[/tex].
First, we evaluate [tex]\( h(0) \)[/tex]. According to the piecewise function definition:
[tex]\[
h(x) =
\begin{cases}
3x - 4, & \text{if } x < 0 \\
2x^2 - 3x + 10, & \text{if } 0 \leq x < 4 \\
2^x, & \text{if } x \geq 4
\end{cases}
\][/tex]
For [tex]\( x = 0 \)[/tex], we use the second case:
[tex]\[
h(0) = 2(0)^2 - 3(0) + 10 = 10
\][/tex]
Next, we evaluate [tex]\( h(4) \)[/tex]. According to the piecewise function definition:
[tex]\[
h(x) =
\begin{cases}
3x - 4, & \text{if } x < 0 \\
2x^2 - 3x + 10, & \text{if } 0 \leq x < 4 \\
2^x, & \text{if } x \geq 4
\end{cases}
\][/tex]
For [tex]\( x = 4 \)[/tex], we use the third case:
[tex]\[
h(4) = 2^4 = 16
\][/tex]
Therefore, the values of the function are:
[tex]\[
\begin{array}{l}
h(0) = 10 \\
h(4) = 16
\end{array}
\][/tex]
