Let's examine the transformation involving the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
### Step-by-Step Solution
1. Understanding the Original Function:
The original function is [tex]\( f(x) = \log{x} \)[/tex]. This is a logarithmic function with base 10 (if not otherwise specified).
2. Understanding the Transformed Function:
The transformed function given is [tex]\( g(x) = \log{x} - 3 \)[/tex].
3. Type of Transformation:
The transformation involves subtracting a constant (in this case, 3) from the original function [tex]\( f(x) \)[/tex].
4. Impact of the Transformation:
- When we subtract a constant from a function, it results in a vertical shift of the graph.
- In this case, subtracting 3 from [tex]\( \log{x} \)[/tex] means that every output value of [tex]\( f(x) \)[/tex] is decreased by 3 units to produce [tex]\( g(x) \)[/tex].
5. Description of the Transformation:
- The graph of the function [tex]\( f(x) = \log{x} \)[/tex] is shifted vertically downward by 3 units.
- This means for every value of [tex]\( x \)[/tex] in the domain of the functions, [tex]\( g(x) \)[/tex] will always be 3 units less than [tex]\( f(x) \)[/tex].
### Conclusion
The type of transformation applied to [tex]\( f(x) = \log{x} \)[/tex] to produce [tex]\( g(x) = \log{x} - 3 \)[/tex] is a vertical shift. Specifically, the graph of [tex]\( f(x) \)[/tex] is shifted downward by 3 units.
This transformation does not affect the domain or the vertical asymptote of the function [tex]\( f(x) \)[/tex]; it only shifts the graph along the vertical axis.
