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:

f(x)=3(14)x5f(x) = 3 \left( \frac{1}{4} \right)^x - 5

Let's analyze the end behavior of this function:

  1. As xx \to -\infty:

    • The term (14)x\left( \frac{1}{4} \right)^x behaves as follows: When xx becomes very negative, (14)x\left( \frac{1}{4} \right)^x increases without bound because 14\frac{1}{4} raised to a large negative number becomes very large (approaching infinity).
    • Therefore, 3(14)x+3 \left( \frac{1}{4} \right)^x \to +\infty.
    • Hence, the function f(x)f(x) approaches ++\infty as xx \to -\infty.
  2. As x+x \to +\infty:

    • The term (14)x\left( \frac{1}{4} \right)^x behaves as follows: As xx becomes very large, (14)x\left( \frac{1}{4} \right)^x approaches 0 because 14\frac{1}{4} raised to a large positive number becomes very small.
    • Therefore, 3(14)x03 \left( \frac{1}{4} \right)^x \to 0.
    • Hence, the function f(x)f(x) approaches 5-5 (because of the constant subtraction of 5).

Conclusion:

  • As xx \to -\infty, f(x)+f(x) \to +\infty.
  • As x+x \to +\infty, f(x)5f(x) \to -5.

So, the correct answer is:

B. As xx \to -\infty, y+y \to +\infty. As x+x \to +\infty, y5y \to -5.


Would you like further explanation or details on the function's behavior? Here are some related questions for you to explore:

  1. How do exponential functions with bases between 0 and 1 behave at extreme values of xx?
  2. What happens to the graph of an exponential function when it's multiplied by a constant factor?
  3. How can you identify the horizontal asymptote of an exponential function?
  4. What does the behavior of a function like (14)x\left( \frac{1}{4} \right)^x tell you about the rates of growth and decay?
  5. How do transformations affect the graph of an exponential function?

Tip: Remember, the horizontal asymptote of an exponential function abx+ca \cdot b^x + c is determined by the value cc, 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