Understanding End Behavior of Functions

TLDR; Understanding end behavior or long run behavior of functions involves determining the value of the function as x approaches positive or negative infinity.

💡 End Behavior Definition

End behavior or long run behavior of a function describes the value of the function or the y value as x approaches positive or negative infinity.

As x approaches positive infinity, the function value approaches some value, and as x approaches negative infinity, the function value also approaches some value.

For instance, as x approaches positive infinity, the y values start to increase without bound, approaching positive infinity.

Similarly, as x approaches negative infinity, the y values start to decrease without bound, approaching negative infinity.

📈 End Behavior as X Approaches Positive Infinity

As x approaches positive infinity, the function moves right along the x-axis, and the y value approaches positive infinity.

Moving right along the graph, the y values start to increase without bound, or the function values increase without bound, approaching positive infinity.

Therefore, as x approaches positive infinity, f of x also approaches infinity.

📉 End Behavior as X Approaches Negative Infinity

As x approaches negative infinity, the function moves left along the x-axis, and the y values or function values are approaching negative infinity.

Moving left along the function approaching x = negative infinity, the graph starts to go down very quickly, causing the y values or function values to approach negative infinity.

Hence, as x approaches negative infinity, f of x approaches negative infinity.

🔍 Second Example of End Behavior

In the second example, as x approaches positive infinity, the y value approaches positive infinity, and as x approaches negative infinity, the y value also approaches positive infinity.

As x approaches positive infinity, moving right along the function causes the y values to increase without bound, approaching positive infinity.

Similarly, as x approaches negative infinity, moving left along the function results in the y values increasing without bound, approaching positive infinity.

🎯 Summary of End Behavior

The two statements in each example describe the end behavior or long run behavior of the given function.

Understanding the end behavior of functions involves determining the value of the function as x approaches positive or negative infinity, providing insights into the behavior of the function at extreme values of x.