Answered

Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

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 :

GinnyAnswer
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]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.