The function $$ g $$ is defined as follows. $$ g(x)=-2 x^{2}+1 $$ If the graph of $$ g $$ is translated vertically upward by 4 units, it becomes the graph of a function $$ f $$. Find the expression for $$ f(x) $$.

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We are given 
$$
g(x) = -2x^2 + 1.
$$
A vertical translation upward by 4 units means we add 4 to the output of $$ g $$. Therefore, the new function $$ f $$ is given by
$$
f(x) = g(x) + 4.
$$

Substitute $$ g(x) $$ into the expression:
$$
f(x) = (-2x^2 + 1) + 4.
$$

Simplify the expression:
$$
f(x) = -2x^2 + 5.
$$
$$ f(x) = -2x^2 + 5 $$

The function is defined as follows.

If the graph of is translated vertically upward by 4 units, it becomes the graph of a function .
Find the expression for .

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The function is defined as follows. If the graph of is translated vertically upward by 4 units, it becomes the graph of a function . Find the expression for .