To find the exponential function that goes through the points [tex]\((0,3)\)[/tex] and [tex]\((1,12)\)[/tex], we begin by considering the general form of an exponential function:
[tex]\[ h(x) = a \cdot b^x \][/tex]
We need to determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] using the given points.
1. Using the point [tex]\((0, 3)\)[/tex]:
Since [tex]\(h(0) = 3\)[/tex]:
[tex]\[
a \cdot b^0 = 3 \implies a \cdot 1 = 3 \implies a = 3
\][/tex]
2. Using the point [tex]\((1, 12)\)[/tex]:
Since [tex]\(h(1) = 12\)[/tex]:
[tex]\[
a \cdot b^1 = 12 \implies 3 \cdot b = 12 \implies b = \frac{12}{3} \implies b = 4
\][/tex]
Now we have the values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ a = 3 \][/tex]
[tex]\[ b = 4 \][/tex]
Thus, the equation of the exponential function is:
[tex]\[ h(x) = 3 \cdot 4^x \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{h(x) = 3(4)^x} \][/tex]
This corresponds to option D.
