Alright, let's carefully analyze the given options for the transformation \( g \) of the rational function \( f \). We are interested in how the transformation affects the behavior of the function as \( x \) approaches infinity or zero.
Given:
- \( g(x) \) is a transformation of \( f(x) \).
- As \( x \) approaches infinity for both functions, \( f(x) \) approaches 0.
We need to identify how each potential equation affects the behavior of \( f \) under the transformation to form \( g \).
We'll consider each option separately:
1. Option A: \( g(x) = f(x+3) - 3 \)
- This means \( g(x) \) is the result of shifting \( f(x) \) 3 units to the left and then subtracting 3 from the result.
- As \( x \) approaches infinity, \( x+3 \) also approaches infinity, so \( f(x+3) \) still approaches 0.
- After shifting, \( f(x+3) \), the transformation \( g(x) = f(x+3) - 3 \) will approach \( 0 - 3 = -3 \), not 0.
2. Option B: \( g(x) = -f(x+3) \)
- This means \( g(x) \) is the result of shifting \( f(x) \) 3 units to the left and then multiplying the result by -1.
- As \( x \) approaches infinity, \( x+3 \) also approaches infinity, so \( f(x+3) \) still approaches 0.
- After shifting, \( f(x+3) \) approaches 0, and the transformation \( g(x) = -f(x+3) \) will approach \(-0 = 0\).
- This transformation does not change the behavior regarding the limit as \( x \) approaches infinity because multiplying 0 by anything remains 0.
3. Option C: \( g(x) = 3f(x) - 3 \)
- This means \( g(x) \) is the result of scaling \( f(x) \) by a factor of 3 and then subtracting 3 from the result.
- As \( x \) approaches infinity, \( f(x) \) approaches 0.
- Multiplying by 3 does not change the limit; it will still be \( 3 \times 0 = 0 \).
- After scaling, the transformation \( g(x) = 3f(x) - 3 \) will approach \( 0 - 3 = -3\), not 0.
4. Option D: \( g(x) = -f(x) + 3 \)
- This means \( g(x) \) is the result of multiplying \( f(x) \) by -1 and then adding 3 to the result.
- As \( x \) approaches infinity, \( f(x) \) approaches 0.
- After multiplying by -1, \( -f(x) \) still approaches \(-0 = 0\), and adding 3 to this result will approach \( 0 + 3 = 3 \), not 0.
Based on the detailed analysis:
- The only equation where \( g(x) \) retains the property \( g(x) \to 0 \) as \( x \to \infty \) is Option B: \( g(x) = -f(x+3) \).
So, the correct answer is:
[tex]\[ g(x) = -f(x+3) \][/tex]
