Choose the values of a and b for which the function f(x) = ab^x meets the given requirements: f(x) → -∞ as x → -∞ and f(x) → 0 as x → +∞.
Understand the Problem
The question is asking for the values of the parameters 'a' and 'b' in the function f(x) = ab^x, given specific behavior as x approaches negative and positive infinity. Specifically, it requires conditions for f(x) to diverge to negative infinity as x approaches negative infinity and to converge to zero as x approaches positive infinity.
Answer
Using \( a < 0 \) and \( 0 < b < 1 \), we find \( f(x) \to -\infty \) as \( x \to -\infty \) and \( f(x) \to 0 \) as \( x \to +\infty \).
Answer for screen readers
The values of ( a ) and ( b ) that meet the requirements are ( a < 0 ) and ( 0 < b < 1 ).
Steps to Solve
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Understanding the limits as ( x ) approaches infinities
We need to analyze the behavior of the function ( f(x) = ab^x ) under two conditions: ( f(x) \to -\infty ) as ( x \to -\infty ) and ( f(x) \to 0 ) as ( x \to +\infty ).
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Condition for ( f(x) \to -\infty )
As ( x \to -\infty ), ( b^x ) must behave in such a way that it diverges to negative infinity. This scenario occurs when ( a < 0 ) and ( 0 < b < 1 ). Here, ( b^x ) approaches ( \infty ) since ( b^x = \frac{1}{(1/b)^{-x}} ), so ( a ) (being negative) causes the product to diverge negatively.
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Condition for ( f(x) \to 0 )
As ( x \to +\infty ), we need ( f(x) = ab^x \to 0 ). This occurs when ( 0 < b < 1 ), which implies that ( b^x \to 0 ) as ( x \to +\infty ). If ( a ) is negative, the product starts to trend toward 0, fulfilling this limit condition.
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Concluding the valid parameters
Thus, ensuring both conditions are valid leads us to conclude that the values ( a < 0 ) and ( 0 < b < 1 ) fulfill both requirements.
The values of ( a ) and ( b ) that meet the requirements are ( a < 0 ) and ( 0 < b < 1 ).
More Information
This configuration ensures that the function ( f(x) = ab^x ) diverges to negative infinity as ( x ) approaches negative infinity, while simultaneously approaching zero as ( x ) approaches positive infinity. This is particularly useful in exponential decay models.
Tips
- Assuming ( b > 1 ) is valid; values greater than 1 lead to ( f(x) \to \infty ) instead of approaching zero as ( x \to \infty ).
- Neglecting the negative sign on ( a ); a positive ( a ) would result in ( f(x) \to \infty ) as ( x \to -\infty ).
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