Select The Correct Answer.Describe The End Behavior Of The Following Exponential Function: $f(x)=3^x$A. As The $x$-values Increase, The Function Increases Towards Positive Infinity.B. As The $x$-values Increase, The Function
**Select the Correct Answer: Describing the End Behavior of Exponential Functions**
What is End Behavior in Exponential Functions?
End behavior in exponential functions refers to the behavior of the function as the input values (x) approach positive or negative infinity. In other words, it describes how the function behaves when the input values are very large or very small.
Q: What is the end behavior of the exponential function f(x) = 3^x?
A: As the x-values increase, the function increases towards positive infinity.
Explanation:
The exponential function f(x) = 3^x is a one-to-one function, which means that it passes the horizontal line test. This means that as the x-values increase, the function values also increase. In fact, the function values increase exponentially, which means that they grow much faster than linear functions.
To understand why the function increases towards positive infinity, let's consider what happens when x is a large positive number. As x gets larger and larger, the function value 3^x gets even larger and larger. In fact, the function value grows so rapidly that it approaches positive infinity as x approaches positive infinity.
Q: What is the end behavior of the exponential function f(x) = 2^(-x)?
A: As the x-values increase, the function approaches 0.
Explanation:
The exponential function f(x) = 2^(-x) is a decreasing function, which means that as the x-values increase, the function values decrease. In fact, the function values decrease exponentially, which means that they decrease much faster than linear functions.
To understand why the function approaches 0, let's consider what happens when x is a large positive number. As x gets larger and larger, the function value 2^(-x) gets smaller and smaller. In fact, the function value approaches 0 as x approaches positive infinity.
Q: What is the end behavior of the exponential function f(x) = e^x?
A: As the x-values increase, the function increases towards positive infinity.
Explanation:
The exponential function f(x) = e^x is a one-to-one function, which means that it passes the horizontal line test. This means that as the x-values increase, the function values also increase. In fact, the function values increase exponentially, which means that they grow much faster than linear functions.
To understand why the function increases towards positive infinity, let's consider what happens when x is a large positive number. As x gets larger and larger, the function value e^x gets even larger and larger. In fact, the function value grows so rapidly that it approaches positive infinity as x approaches positive infinity.
Q: What is the end behavior of the exponential function f(x) = e^(-x)?
A: As the x-values increase, the function approaches 0.
Explanation:
The exponential function f(x) = e^(-x) is a decreasing function, which means that as the x-values increase, the function values decrease. In fact, the function values decrease exponentially, which means that they decrease much faster than linear functions.
To understand why the function approaches 0, let's consider what happens when x is a large positive number. As x gets larger and larger, the function value e^(-x) gets smaller and smaller. In fact, the function value approaches 0 as x approaches positive infinity.
Conclusion:
In conclusion, the end behavior of exponential functions depends on the base of the function. If the base is greater than 1, the function increases towards positive infinity as the x-values increase. If the base is between 0 and 1, the function approaches 0 as the x-values increase.