Answered

IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.

Which of the following functions would result in a graph that is shifted two units to the left of [tex]g(x)=3 \log (x)[/tex]?

A. [tex]f(x)=3 \log (x)+2[/tex]
B. [tex]f(x)=3 \log (x+2)[/tex]
C. [tex]f(x)=3 \log (x)-2[/tex]
D. [tex]f(x)=3 \log (x-2)[/tex]

Sagot :

GinnyAnsweridn
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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.