Select the correct answer from each drop-down menu.
Complete the statement to correctly describe the end behavior of the given function.
[tex]f(x)=10(0.75)^x[/tex]
The left end approaches [ ] , and the right end approaches [ ]
Answer (1)
The function is f ( x ) = 10 ( 0.75 ) x .
As x approaches positive infinity, f ( x ) approaches 0.
As x approaches negative infinity, f ( x ) approaches infinity.
The left end approaches ∞ , and the right end approaches 0 . Therefore, the answer is 0 .
Explanation
Understanding the Problem We are given the function f ( x ) = 10 ( 0.75 ) x and we need to describe its end behavior. This means we need to determine what happens to the function as x approaches positive infinity and as x approaches negative infinity.
Analyzing the Right End Behavior As x approaches positive infinity ( x → ∞ ), the term ( 0.75 ) x approaches 0 because 0.75 is a fraction between 0 and 1. Therefore, f ( x ) = 10 ( 0.75 ) x approaches 10 × 0 = 0 .
Analyzing the Left End Behavior As x approaches negative infinity ( x → − ∞ ), the term ( 0.75 ) x approaches infinity. This is because ( 0.75 ) x = ( 4 3 ) x = ( 3 4 ) − x . As x becomes a large negative number, − x becomes a large positive number, and ( 3 4 ) − x grows without bound. Therefore, f ( x ) = 10 ( 0.75 ) x approaches infinity.
Conclusion In summary, as x goes to positive infinity, f ( x ) approaches 0, and as x goes to negative infinity, f ( x ) approaches infinity. Therefore, the left end approaches ∞ and the right end approaches 0.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a savings account with a fixed interest rate, the amount of money you have will grow exponentially over time. Similarly, the decay of a radioactive substance can be modeled using an exponential function. Understanding the end behavior of exponential functions helps us predict long-term trends in these scenarios.
