(A) as \( x \rightarrow-\infty, f(x) \rightarrow-\infty \) and as \( x \rightarrow \infty, f(x) \rightarrow-\infty \) (B) as \( x \rightarrow-\infty, f(x) \rightarrow \infty \) and as \( x \rightarrow \infty, f(x) \rightarrow-\infty \)

The behavior of a function \( f(x) \) as \( x \) approaches \( -\infty \) and \( \infty \) can provide important information about its overall shape and characteristics. Let's analyze the two cases you've presented: ### Case (A) - As \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \) - As \( x \rightarrow \infty \), \( f(x) \rightarrow -\infty \) In this case, the function \( f(x) \) decreases without bound as \( x \) moves towards both negative and positive infinity. This suggests that the function may have a maximum value at some finite point, but it does not rise to positive values as \( x \) goes to either extreme. A common example of such a function is a downward-opening parabola, like \( f(x) = -x^2 \). ### Case (B) - As \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \) - As \( x \rightarrow \infty \), \( f(x) \rightarrow -\infty \) In this scenario, the function \( f(x) \) increases without bound as \( x \) approaches negative infinity, but decreases without bound as \( x \) approaches positive infinity. This indicates that the function has a maximum point somewhere in between, and it could resemble an upward-opening parabola that eventually decreases, such as \( f(x) = x^2 - 1 \). ### Summary - **Case (A)** indicates a function that is always decreasing and approaches negative infinity at both extremes. - **Case (B)** indicates a function that increases to positive infinity on the left and decreases to negative infinity on the right, suggesting a maximum point in between. If you have a specific function in mind or need further analysis, feel free to provide more details!