To determine which function graph passes through the point [tex]\((0, 4)\)[/tex], we need to evaluate each function at [tex]\( x = 0 \)[/tex] and check which one gives the value [tex]\( 4 \)[/tex].
Let's evaluate each function at [tex]\( x = 0 \)[/tex]:
1. [tex]\( f(x) = \sin(x) + 4 \)[/tex]
[tex]\[
f(0) = \sin(0) + 4 = 0 + 4 = 4
\][/tex]
So, this function passes through [tex]\((0, 4)\)[/tex].
2. [tex]\( f(x) = \cos(x) + 3 \)[/tex]
[tex]\[
f(0) = \cos(0) + 3 = 1 + 3 = 4
\][/tex]
So, this function also passes through [tex]\((0, 4)\)[/tex].
3. [tex]\( f(x) = -3 \sin(x) \)[/tex]
[tex]\[
f(0) = -3 \sin(0) = -3 \cdot 0 = 0
\][/tex]
This function does not pass through [tex]\((0, 4)\)[/tex].
4. [tex]\( f(x) = 4 \cos(x) \)[/tex]
[tex]\[
f(0) = 4 \cos(0) = 4 \cdot 1 = 4
\][/tex]
So, this function also passes through [tex]\((0, 4)\)[/tex].
From the evaluations, we see that the functions [tex]\( f(x) = \sin(x) + 4 \)[/tex], [tex]\( f(x) = \cos(x) + 3 \)[/tex], and [tex]\( f(x) = 4 \cos(x) \)[/tex] all pass through the point [tex]\((0, 4)\)[/tex]. However, the index corresponding to these functions are ordered as: [tex]\( f(x) = \sin (x) + 4 \)[/tex], [tex]\( f(x) = \cos(x) + 3 \)[/tex], [tex]\( f(x) = 4 \cos(x) \)[/tex] in the list (from options starting from 1).
Hence, the function [tex]\( f(x) = \sin(x) + 4 \)[/tex] (the one at index 1) is the one that passes through [tex]\((0, 4)\)[/tex] and meets the required criteria.
