Math Problem Statement
Consider the following function:
f(x)= 3 ( 1 4 ) x − 5
Which statement is true about the functions endbehavior?
A.
As x -> − ∞ , then y -> − ∞ .As x -> ∞ , then y -> -5.
B.
As x -> − ∞ , then y -> ∞ .As x -> ∞ , then y -> -5.
C.
As x -> − ∞ , then y -> -5. As x -> ∞ , then y -> ∞ .
D.
As x -> − ∞ , then y -> -5.As x -> ∞ , then y -> − ∞ .
Solution
The function given is:
Let's analyze the end behavior of this function:
-
As :
- The term behaves as follows: When becomes very negative, increases without bound because raised to a large negative number becomes very large (approaching infinity).
- Therefore, .
- Hence, the function approaches as .
-
As :
- The term behaves as follows: As becomes very large, approaches 0 because raised to a large positive number becomes very small.
- Therefore, .
- Hence, the function approaches (because of the constant subtraction of 5).
Conclusion:
- As , .
- As , .
So, the correct answer is:
B. As , . As , .
Would you like further explanation or details on the function's behavior? Here are some related questions for you to explore:
- How do exponential functions with bases between 0 and 1 behave at extreme values of ?
- What happens to the graph of an exponential function when it's multiplied by a constant factor?
- How can you identify the horizontal asymptote of an exponential function?
- What does the behavior of a function like tell you about the rates of growth and decay?
- How do transformations affect the graph of an exponential function?
Tip: Remember, the horizontal asymptote of an exponential function is determined by the value , which is the vertical shift.
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Math Problem Analysis
Mathematical Concepts
End Behavior of Functions
Exponential Functions
Asymptotes
Formulas
f(x) = 3(1/4)^x - 5
Theorems
End Behavior of Exponential Functions
Horizontal Asymptotes of Exponential Functions
Suitable Grade Level
Grades 9-11
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