Select the correct answer.
Which function is decreasing and approaches negative infinity as $x$ increases?
A. $f(x)=3(0.6)^x-1$
B. $f(x)=-3(0.6)^x+1$
C. $f(x)=3(6)^x-2$
D. $f(x)=-3(6)^x+2
Answer (1)
Analyze the properties of exponential functions of the form f ( x ) = a x + c based on the values of a and b .
Determine that for a function to be decreasing and approach negative infinity as x increases, we need a < 0 and 1"> b > 1 .
Apply these rules to the given functions.
Conclude that the function f ( x ) = − 3 ( 6 ) x + 2 satisfies the condition, so the answer is D .
Explanation
Understanding the Problem We are given four functions and we need to determine which one is decreasing and approaches negative infinity as x increases.
Listing the Functions A. f ( x ) = 3 ( 0.6 ) x − 1 B. f ( x ) = − 3 ( 0.6 ) x + 1 C. f ( x ) = 3 ( 6 ) x − 2 D. f ( x ) = − 3 ( 6 ) x + 2
Analyzing Exponential Functions We need to find a function that is decreasing and approaches negative infinity as x increases. Let's analyze the general form of exponential functions.
Rules for Exponential Functions For a function of the form f ( x ) = a ⋅ b x + c , where a is a constant, b is the base, and c is a constant:
If 0"> a > 0 and 0 < b < 1 , the function is decreasing and approaches c as x approaches infinity.
If a < 0 and 0 < b < 1 , the function is increasing and approaches c as x approaches infinity.
If 0"> a > 0 and 1"> b > 1 , the function is increasing and approaches infinity as x approaches infinity.
If a < 0 and 1"> b > 1 , the function is decreasing and approaches negative infinity as x approaches infinity.
Applying the Rules to the Given Functions Now, let's apply these rules to each of the given functions:
A. f ( x ) = 3 ( 0.6 ) x − 1 . Here, 0"> a = 3 > 0 and 0 < b = 0.6 < 1 . This function is decreasing and approaches − 1 as x approaches infinity.
B. f ( x ) = − 3 ( 0.6 ) x + 1 . Here, a = − 3 < 0 and 0 < b = 0.6 < 1 . This function is increasing and approaches 1 as x approaches infinity.
C. f ( x ) = 3 ( 6 ) x − 2 . Here, 0"> a = 3 > 0 and 1"> b = 6 > 1 . This function is increasing and approaches infinity as x approaches infinity.
D. f ( x ) = − 3 ( 6 ) x + 2 . Here, a = − 3 < 0 and 1"> b = 6 > 1 . This function is decreasing and approaches negative infinity as x approaches infinity.
Finding the Correct Function We are looking for a function that is decreasing and approaches negative infinity as x increases. Based on our analysis, function D satisfies this condition.
Final Answer Therefore, the correct answer is D.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding how the parameters of an exponential function affect its behavior is crucial in making predictions and informed decisions in these areas. For example, in finance, understanding exponential decay can help estimate the depreciation of an asset over time, while in biology, it can model the decay of a drug in the body.
