Which Statement About The End Behavior Of The Logarithmic Function F(X) = Log(X + 3) – 2 Is True?

If you are studying the end behavior of the logarithmic function f(x) = log(x + 3) – 2, then you might be wondering which statement about its end behavior is true. Logarithmic functions can be tricky to analyze due to its inverse nature, but understanding it is key for graphing and solving equations. In this article, we will take a closer look at the end behavior of f(x) = log(x + 3) – 2 by examining what happens as x approaches infinity or negative infinity. By the end, you should have a better understanding of how to determine the end behavior of a given logarithmic function.

The nature of the end behavior of a logarithmic function

When graphed, the logarithmic function f(x) = log(x + ) has asymptotes at x = – and x = . The function approaches these asymptotes as x gets large in either direction.

The end behavior of the logarithmic function f(x) = log(x + 3) – 2

When x approaches infinity, the logarithmic function f(x) = log(x + 3) – 2 approaches 0. When x approaches negative infinity, the function also approaches 0. This can be seen by taking the limit as x approaches infinity and negative infinity of the logarithmic function f(x) = log(x + 3) – 2.

Why the end behavior of a logarithmic function is important

The end behavior of the logarithmic function F(X) = Log(X + ) – is important because it describes how the function behaves as X approaches infinity or negative infinity. In particular, the end behavior can be used to determine whether the function is bounded or unbounded.

How to use the end behavior of a logarithmic function to solve problems

As we know, the end behavior of a logarithmic function is determined by its leading term. In this case, the leading term is . Therefore, the end behavior of is to approach infinity as x approaches infinity, and to approach negative infinity as x approaches 0 from the right.

We can use this information to solve problems involving limits at infinity and horizontal asymptotes. For example, consider the following problem:

Find the limit of as x approaches infinity.

Since the leading term of is , we know that will approach infinity as x approaches infinity. Therefore, we can say that the limit of as x approaches infinity is infinity.

Now let’s look at an example involving a horizontal asymptote. Consider the following function:

The leading term of is , so we know that will approach negative infinity as x approaches 0 from the right. This means that the function has a horizontal asymptote at y = -∞.

Conclusion

In conclusion, the end behavior of the logarithmic function f(x) = log(x + 3) – 2 is positive infinity when x approaches negative infinity and negative infinity when x approaches positive infinity. This is due to the fact that as x increases, the value inside the brackets (x + 3) becomes larger and larger, thus resulting in a higher logarithm result which tends toward positive or negative infinity depending on its sign. As such, it can be concluded that this type of function has an infinite range at both ends of its domain.