To determine the interval over which the function f(g(x)) is negative, we first need to analyze the behaviors of both f and g.
1. **Understanding the Functions**: Begin by examining the function g(x). Identify its range and any important points (like intercepts or asymptotes). This will help you understand what output values g(x) can take as x varies.
2. **Applying the Inner Function**: Once you have a grasp on g(x), substitute it into f. The next step is to identify where the output of f(g(x)) is less than zero. This typically requires looking for the values of g(x) that result in negative outputs when plugged into f.
3. **Finding Critical Points**: Solve the equation f(g(x)) = 0 to find critical points. These are the x-values where f(g(x)) transitions from positive to negative or vice versa. To find out where f(g(x)) is negative, assess the sign of f(g(x)) in intervals defined by these critical points.
4. **Testing Intervals**: For each interval created by the critical points, choose a test point and plug it into the function f(g(x)). If the output is negative in that interval, then f(g(x)) is negative in that segment.
5. **Conclusion**: Finally, summarize your findings by stating the interval(s) in which f(g(x)) is negative. For example, you might conclude that it is negative for x>a and x<b where a and b are determined critical points.
In essence, navigating through the interplay between f and g allows us to pinpoint the specifics of where their composite function f(g(x)) takes on negative values. The precise intervals will depend on the defined forms of f and g, so ensure you assess them carefully!