To determine the effect on the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] when it is transformed to [tex]\( g(x) = \frac{1}{x} - 14 \)[/tex], we need to analyze the transformation applied to the function.
The given transformation is:
[tex]\[ g(x) = \frac{1}{x} - 14 \][/tex]
Here, the function [tex]\( g(x) \)[/tex] is obtained by subtracting 14 from the original function [tex]\( f(x) \)[/tex]. This form [tex]\( g(x) = f(x) - k \)[/tex] indicates a vertical shift of the original graph of [tex]\( f(x) \)[/tex].
- When [tex]\( k \)[/tex] is positive, [tex]\( f(x) \)[/tex] is shifted [tex]\( k \)[/tex] units downward.
- When [tex]\( k \)[/tex] is negative, [tex]\( f(x) \)[/tex] is shifted [tex]\( |k| \)[/tex] units upward.
In our case, [tex]\( -14 \)[/tex] signifies a downward shift by 14 units.
Therefore, the correct answer is:
D. The graph of [tex]\( f(x) \)[/tex] is shifted 14 units down.
