To solve this problem, we need to analyze the given function [tex]\( f(x) = \log(x) \)[/tex] and identify the modifications provided in each of the choices. We will determine which expression corresponds to the function [tex]\( g(x) \)[/tex].
The choices given are:
A. [tex]\(\log(x) - 2\)[/tex]
B. [tex]\(\log(x + 2)\)[/tex]
C. [tex]\(\log(x) + 2\)[/tex]
D. [tex]\(\log(x - 2)\)[/tex]
Let us review each choice thoughtfully:
1. Choice A: [tex]\(\log(x) - 2\)[/tex]
- This expression represents the logarithm of [tex]\( x \)[/tex] subtracted by 2.
- Essentially, if [tex]\( g(x) = \log(x) - 2 \)[/tex], then [tex]\( g(x) = f(x) - 2 \)[/tex].
2. Choice B: [tex]\(\log(x + 2)\)[/tex]
- This expression takes the logarithm of [tex]\( x \)[/tex] after adding 2 to [tex]\( x \)[/tex].
- So, if [tex]\( g(x) = \log(x + 2) \)[/tex], [tex]\( g(x) = \log(x + 2) \)[/tex] changes the input to the logarithmic function.
3. Choice C: [tex]\(\log(x) + 2\)[/tex]
- This expression represents the logarithm of [tex]\( x \)[/tex] with 2 added to the result of the logarithm.
- Therefore, if [tex]\( g(x) = \log(x) + 2 \)[/tex], [tex]\( g(x) = f(x) + 2 \)[/tex].
4. Choice D: [tex]\(\log(x - 2)\)[/tex]
- This expression takes the logarithm of [tex]\( x \)[/tex] after subtracting 2 from [tex]\( x \)[/tex].
- Hence, if [tex]\( g(x) = \log(x - 2) \)[/tex], [tex]\( g(x) = \log(x - 2) \)[/tex] changes the input to the logarithmic function.
In summary:
- A. [tex]\(\log(x) - 2\)[/tex] corresponds to [tex]\( f(x) - 2 \)[/tex].
- B. [tex]\(\log(x + 2)\)[/tex] corresponds to changing the input to [tex]\( f(x) \)[/tex] by adding 2.
- C. [tex]\(\log(x) + 2\)[/tex] corresponds to [tex]\( f(x) + 2 \)[/tex].
- D. [tex]\(\log(x - 2)\)[/tex] corresponds to changing the input to [tex]\( f(x) \)[/tex] by subtracting 2.
So, the expressions found for [tex]\( g(x) \)[/tex] in terms of [tex]\( f(x) \)[/tex] are:
- [tex]\(\log(x) - 2\)[/tex]
- [tex]\(\log(x + 2)\)[/tex]
- [tex]\(\log(x) + 2\)[/tex]
- [tex]\(\log(x - 2)\)[/tex]
These modifications reflect all given choices, showing different ways of manipulating the basic function [tex]\( f(x) = \log(x) \)[/tex] to form [tex]\( g(x) \)[/tex].
