To solve for [tex]\( g(x) \)[/tex] in the simplest form, follow these steps:
1. Identify the original function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 3x^2 + 5
\][/tex]
2. Understand the transformation:
Translating a function upward by a certain number of units means you add that number to the original function. In this case, we are translating the function [tex]\( f(x) \)[/tex] 3 units up.
3. Apply the translation:
Since we are translating [tex]\( f(x) \)[/tex] three units upward, we need to add 3 to [tex]\( f(x) \)[/tex]:
[tex]\[
g(x) = f(x) + 3
\][/tex]
4. Substitute [tex]\( f(x) \)[/tex] into the equation:
[tex]\[
g(x) = (3x^2 + 5) + 3
\][/tex]
5. Combine like terms:
Simplify the right-hand side by combining the constant terms (5 and 3):
[tex]\[
g(x) = 3x^2 + 5 + 3 = 3x^2 + 8
\][/tex]
Therefore, the equation for [tex]\( g(x) \)[/tex] in its simplest form is:
[tex]\[
g(x) = 3x^2 + 8
\][/tex]
