To determine the values of the function [tex]\( h \)[/tex] at [tex]\( x = 0 \)[/tex] and [tex]\( x = 4 \)[/tex], we must carefully consider the given piecewise function and evaluate it at each specific point.
The definition of the function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x)=\left\{\begin{array}{ll}
3 x - 4, & x < 0 \\
2 x^2 - 3 x + 10, & 0 \leq x < 4 \\
2^x, & x \geq 4
\end{array}\right.
\][/tex]
Step 1: Calculate [tex]\( h(0) \)[/tex]
Since [tex]\( 0 \leq 0 < 4 \)[/tex], we use the second piece of the piecewise function:
[tex]\[
h(x) = 2x^2 - 3x + 10, \quad 0 \leq x < 4.
\][/tex]
Substitute [tex]\( x = 0 \)[/tex] into this expression:
[tex]\[
h(0) = 2(0)^2 - 3(0) + 10 = 0 - 0 + 10 = 10.
\][/tex]
Thus,
[tex]\[
h(0) = 10.
\][/tex]
Step 2: Calculate [tex]\( h(4) \)[/tex]
Since [tex]\( x = 4 \geq 4 \)[/tex], we use the third piece of the piecewise function:
[tex]\[
h(x) = 2^x, \quad x \geq 4.
\][/tex]
Substitute [tex]\( x = 4 \)[/tex] into this expression:
[tex]\[
h(4) = 2^4 = 16.
\][/tex]
Thus,
[tex]\[
h(4) = 16.
\][/tex]
Summary of Results:
[tex]\[
\begin{array}{l}
h(0) = 10 \\
h(4) = 16
\end{array}
\][/tex]
