Answer :
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].
