Let's analyze the given function transformation step-by-step to determine the correct statement about the effects on the graph of function [tex]\( f \)[/tex] to obtain the graph of [tex]\( g \)[/tex].
The given transformation is [tex]\( g(x) = f(x - 3) + 4 \)[/tex].
### Step 1: Horizontal Shift
The term [tex]\( x - 3 \)[/tex] inside the function [tex]\( f \)[/tex]:
1. The expression [tex]\( f(x - h) \)[/tex] indicates a horizontal shift of the graph of [tex]\( f(x) \)[/tex].
2. If [tex]\( h \)[/tex] is positive (i.e., [tex]\( f(x - 3) \)[/tex]), it means the graph of [tex]\( f \)[/tex] is shifted to the right by [tex]\( h \)[/tex] units.
3. Thus, [tex]\( x - 3 \)[/tex] represents a horizontal shift to the right by 3 units.
### Step 2: Vertical Shift
The term [tex]\( + 4 \)[/tex] outside the function [tex]\( f \)[/tex]:
1. The expression [tex]\( f(x) + k \)[/tex] indicates a vertical shift of the graph of [tex]\( f(x) \)[/tex].
2. If [tex]\( k \)[/tex] is positive (i.e., [tex]\( +4 \)[/tex]), it means the graph of [tex]\( f \)[/tex] is shifted upwards by [tex]\( k \)[/tex] units.
3. Thus, [tex]\( +4 \)[/tex] represents a vertical shift upwards by 4 units.
### Conclusion
Combining both transformations:
- The graph of the function [tex]\( f \)[/tex] is shifted to the right by 3 units.
- The graph of the function [tex]\( f \)[/tex] is shifted up by 4 units.
Therefore, the correct statement describing the transformation is:
A. The graph of function [tex]\( f \)[/tex] is shifted right 3 units and up 4 units.
This is the true statement based on the given transformations.
