Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To determine whether [tex]\( y = 0 \)[/tex] is the horizontal asymptote for all functions of the form [tex]\( f(x) = ab^x \)[/tex], we need to analyze the behavior of this function as [tex]\( x \)[/tex] approaches both positive and negative infinity, assuming [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants with [tex]\( a \neq 0 \)[/tex] and [tex]\( b > 0 \)[/tex].
### Case 1: [tex]\( 0 < b < 1 \)[/tex]
1. As [tex]\( x \to +\infty \)[/tex]:
- When [tex]\( b \)[/tex] is between 0 and 1, raising [tex]\( b \)[/tex] to a large positive power will make [tex]\( b^x \)[/tex] approach 0.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will approach [tex]\( a \cdot 0 = 0 \)[/tex].
- Thus, as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
2. As [tex]\( x \to -\infty \)[/tex]:
- Raising [tex]\( b \)[/tex] to a large negative power (since [tex]\( 0 < b < 1 \)[/tex]) results in [tex]\( b^x \)[/tex] approaching infinity.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will grow very large but since [tex]\( b \)[/tex] is a fraction less than 1, as [tex]\( x \to -\infty \)[/tex], [tex]\( b^x \)[/tex] approaches infinity.
- However, [tex]\( f(x) = a \cdot \infty \)[/tex] if [tex]\( a \neq 0 \)[/tex], which means [tex]\( f(x) \to 0 \)[/tex].
### Case 2: [tex]\( b > 1 \)[/tex]
1. As [tex]\( x \to +\infty \)[/tex]:
- When [tex]\( b \)[/tex] is greater than 1, raising [tex]\( b \)[/tex] to a large positive power will cause [tex]\( b^x \)[/tex] to grow very large.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will also grow very large, and [tex]\( y = 0 \)[/tex] is not an asymptote in this case.
2. As [tex]\( x \to -\infty \)[/tex]:
- Raising [tex]\( b \)[/tex] to a large negative power (since [tex]\( b > 1 \)[/tex]) results in [tex]\( b^x \)[/tex] approaching 0.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will approach [tex]\( a \cdot 0 = 0 \)[/tex].
- Thus, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
### Special Case: [tex]\( a = 0 \)[/tex]
- If [tex]\( a = 0 \)[/tex], then regardless of [tex]\( b \)[/tex], the function [tex]\( f(x) = 0 \)[/tex] is a constant function.
- The graph of [tex]\( f(x) \)[/tex] is a horizontal line at [tex]\( y = 0 \)[/tex], which coincides perfectly with the asymptote [tex]\( y = 0 \)[/tex].
### Conclusion
Based on the above analysis:
- For [tex]\( 0 < b < 1 \)[/tex], both as [tex]\( x \to +\infty \)[/tex] and [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches 0.
- For [tex]\( b > 1 \)[/tex], as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches 0.
- For [tex]\( a = 0 \)[/tex], the function is constantly 0.
Hence, [tex]\( y = 0 \)[/tex] is indeed the horizontal asymptote for all functions of the form [tex]\( f(x) = ab^x \)[/tex].
### Case 1: [tex]\( 0 < b < 1 \)[/tex]
1. As [tex]\( x \to +\infty \)[/tex]:
- When [tex]\( b \)[/tex] is between 0 and 1, raising [tex]\( b \)[/tex] to a large positive power will make [tex]\( b^x \)[/tex] approach 0.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will approach [tex]\( a \cdot 0 = 0 \)[/tex].
- Thus, as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
2. As [tex]\( x \to -\infty \)[/tex]:
- Raising [tex]\( b \)[/tex] to a large negative power (since [tex]\( 0 < b < 1 \)[/tex]) results in [tex]\( b^x \)[/tex] approaching infinity.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will grow very large but since [tex]\( b \)[/tex] is a fraction less than 1, as [tex]\( x \to -\infty \)[/tex], [tex]\( b^x \)[/tex] approaches infinity.
- However, [tex]\( f(x) = a \cdot \infty \)[/tex] if [tex]\( a \neq 0 \)[/tex], which means [tex]\( f(x) \to 0 \)[/tex].
### Case 2: [tex]\( b > 1 \)[/tex]
1. As [tex]\( x \to +\infty \)[/tex]:
- When [tex]\( b \)[/tex] is greater than 1, raising [tex]\( b \)[/tex] to a large positive power will cause [tex]\( b^x \)[/tex] to grow very large.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will also grow very large, and [tex]\( y = 0 \)[/tex] is not an asymptote in this case.
2. As [tex]\( x \to -\infty \)[/tex]:
- Raising [tex]\( b \)[/tex] to a large negative power (since [tex]\( b > 1 \)[/tex]) results in [tex]\( b^x \)[/tex] approaching 0.
- Therefore, [tex]\( f(x) = ab^x \)[/tex] will approach [tex]\( a \cdot 0 = 0 \)[/tex].
- Thus, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
### Special Case: [tex]\( a = 0 \)[/tex]
- If [tex]\( a = 0 \)[/tex], then regardless of [tex]\( b \)[/tex], the function [tex]\( f(x) = 0 \)[/tex] is a constant function.
- The graph of [tex]\( f(x) \)[/tex] is a horizontal line at [tex]\( y = 0 \)[/tex], which coincides perfectly with the asymptote [tex]\( y = 0 \)[/tex].
### Conclusion
Based on the above analysis:
- For [tex]\( 0 < b < 1 \)[/tex], both as [tex]\( x \to +\infty \)[/tex] and [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches 0.
- For [tex]\( b > 1 \)[/tex], as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches 0.
- For [tex]\( a = 0 \)[/tex], the function is constantly 0.
Hence, [tex]\( y = 0 \)[/tex] is indeed the horizontal asymptote for all functions of the form [tex]\( f(x) = ab^x \)[/tex].
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
