To determine the values of the piecewise function [tex]\( h(x) \)[/tex] for [tex]\( x = 0 \)[/tex] and [tex]\( x = 4 \)[/tex], we will evaluate the function as follows:
1. Evaluating [tex]\( h(0) \)[/tex]:
According to the given piecewise function:
[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 piece [tex]\( 2x^2 - 3x + 10 \)[/tex] because [tex]\( 0 \leq x < 4 \)[/tex].
[tex]\[
h(0) = 2(0)^2 - 3(0) + 10 = 0 - 0 + 10 = 10
\][/tex]
So, [tex]\( h(0) = 10 \)[/tex].
2. Evaluating [tex]\( h(4) \)[/tex]:
For [tex]\( x = 4 \)[/tex], we use the piece [tex]\( 2^x \)[/tex] as [tex]\( x \geq 4 \)[/tex].
[tex]\[
h(4) = 2^4 = 16
\][/tex]
So, [tex]\( h(4) = 16 \)[/tex].
Putting these together, we have:
[tex]\[
\begin{array}{l}
h(0) = 10 \\
h(4) = 16
\end{array}
\][/tex]
Thus, the values of the function are:
[tex]\[
h(0) = 10 \\
h(4) = 16
\][/tex]
