Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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]
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.