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To determine which function results in a graph that is shifted two units to the left of [tex]\( g(x) = 3 \log (x) \)[/tex], we need to understand how different transformations affect the graph of a function.
### Analyzing the Transformations
1. Vertical Shifts:
- Adding a constant [tex]\( k \)[/tex] to [tex]\( f(x) \)[/tex]: [tex]\( f(x) + k \)[/tex]
- This shifts the graph up by [tex]\( k \)[/tex] units.
- Subtracting a constant [tex]\( k \)[/tex] from [tex]\( f(x) \)[/tex]: [tex]\( f(x) - k \)[/tex]
- This shifts the graph down by [tex]\( k \)[/tex] units.
2. Horizontal Shifts:
- Adding a constant [tex]\( h \)[/tex] to [tex]\( x \)[/tex]: [tex]\( f(x + h) \)[/tex]
- This shifts the graph to the left by [tex]\( h \)[/tex] units.
- Subtracting a constant [tex]\( h \)[/tex] from [tex]\( x \)[/tex]: [tex]\( f(x - h) \)[/tex]
- This shifts the graph to the right by [tex]\( h \)[/tex] units.
Given these transformations, let's analyze each option one by one:
#### A. [tex]\( f(x) = 3 \log (x) + 2 \)[/tex]
- This equation adds 2 to the output [tex]\( y \)[/tex]-values of [tex]\( g(x) \)[/tex]. It results in a vertical shift of the graph 2 units up.
- This is not the correct transformation for a horizontal shift to the left.
#### B. [tex]\( f(x) = 3 \log (x + 2) \)[/tex]
- This equation adds 2 to the input [tex]\( x \)[/tex]-values. According to the transformation rules, this shifts the graph to the left by 2 units.
- This is the correct transformation for a horizontal shift to the left.
#### C. [tex]\( f(x) = 3 \log (x) - 2 \)[/tex]
- This equation subtracts 2 from the output [tex]\( y \)[/tex]-values of [tex]\( g(x) \)[/tex]. It results in a vertical shift of the graph 2 units down.
- This is not the correct transformation for a horizontal shift to the left.
#### D. [tex]\( f(x) = 3 \log (x - 2) \)[/tex]
- This equation subtracts 2 from the input [tex]\( x \)[/tex]-values. According to the transformation rules, this shifts the graph to the right by 2 units.
- This is not the correct transformation for a horizontal shift to the left.
### Conclusion
Having analyzed all the options, the function that results in a graph that is shifted two units to the left of [tex]\( g(x) = 3 \log (x) \)[/tex] is:
[tex]\[ \boxed{B} \ f(x) = 3 \log (x + 2) \][/tex]
### Analyzing the Transformations
1. Vertical Shifts:
- Adding a constant [tex]\( k \)[/tex] to [tex]\( f(x) \)[/tex]: [tex]\( f(x) + k \)[/tex]
- This shifts the graph up by [tex]\( k \)[/tex] units.
- Subtracting a constant [tex]\( k \)[/tex] from [tex]\( f(x) \)[/tex]: [tex]\( f(x) - k \)[/tex]
- This shifts the graph down by [tex]\( k \)[/tex] units.
2. Horizontal Shifts:
- Adding a constant [tex]\( h \)[/tex] to [tex]\( x \)[/tex]: [tex]\( f(x + h) \)[/tex]
- This shifts the graph to the left by [tex]\( h \)[/tex] units.
- Subtracting a constant [tex]\( h \)[/tex] from [tex]\( x \)[/tex]: [tex]\( f(x - h) \)[/tex]
- This shifts the graph to the right by [tex]\( h \)[/tex] units.
Given these transformations, let's analyze each option one by one:
#### A. [tex]\( f(x) = 3 \log (x) + 2 \)[/tex]
- This equation adds 2 to the output [tex]\( y \)[/tex]-values of [tex]\( g(x) \)[/tex]. It results in a vertical shift of the graph 2 units up.
- This is not the correct transformation for a horizontal shift to the left.
#### B. [tex]\( f(x) = 3 \log (x + 2) \)[/tex]
- This equation adds 2 to the input [tex]\( x \)[/tex]-values. According to the transformation rules, this shifts the graph to the left by 2 units.
- This is the correct transformation for a horizontal shift to the left.
#### C. [tex]\( f(x) = 3 \log (x) - 2 \)[/tex]
- This equation subtracts 2 from the output [tex]\( y \)[/tex]-values of [tex]\( g(x) \)[/tex]. It results in a vertical shift of the graph 2 units down.
- This is not the correct transformation for a horizontal shift to the left.
#### D. [tex]\( f(x) = 3 \log (x - 2) \)[/tex]
- This equation subtracts 2 from the input [tex]\( x \)[/tex]-values. According to the transformation rules, this shifts the graph to the right by 2 units.
- This is not the correct transformation for a horizontal shift to the left.
### Conclusion
Having analyzed all the options, the function that results in a graph that is shifted two units to the left of [tex]\( g(x) = 3 \log (x) \)[/tex] is:
[tex]\[ \boxed{B} \ f(x) = 3 \log (x + 2) \][/tex]
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