Solving and Plotting F(x) |x - 1| - 3|x^2| Through Piecewise Functions

Introduction

The function F(x) |x - 1| - 3|x^2| can be quite complex to deal with because of the modulus functions involved. The goal is to solve and plot this function by removing the modulus, converting it into a piecewise function, and understanding where it is continuous.

Step 1: Identifying the Critical Points

The two critical points given are x 1 and x -frac{2}{3}. These points are where the behavior of the modulus functions changes.

1. When x , then x - 1 and 3x^2 > 0. Therefore, |x - 1| -x 1 and |3x^2| 3x^2.

2. When -frac{2}{3} leq x , then x - 1 and 3x^2 > 0. Therefore, |x - 1| -x 1 and |3x^2| 3x^2.

3. When x geq 1, then x - 1 geq 0 and 3x^2 > 0. Therefore, |x - 1| x - 1 and |3x^2| 3x^2.

Step 2: Defining the Piecewise Function

Using the critical points, we can define the function F(x) in three different intervals:

For x , F(x) -x - 1 - 3x^2 -3x^2 - x - 1. For -frac{2}{3} leq x , F(x) -x - 1 - 3x^2 -3x^2 - x - 1. For x geq 1, F(x) x - 1 - 3x^2 -3x^2 x - 1.

Combining these, we get the piecewise function:

[begin{cases} -3x^2 - x - 1 text{if } x

Step 3: Verifying Continuity

To ensure the function is continuous, we need to check the values at the critical points x frac{2}{3} and x 1.

At x frac{2}{3} Left-hand limit at x frac{2}{3}, Fleft(frac{2}{3}right) -4 cdot frac{2}{3} - 1 -frac{8}{3} - 1 -frac{11}{3} Right-hand limit at x frac{2}{3}, Fleft(frac{2}{3}right) 2 cdot frac{2}{3}^3 2 cdot frac{8}{27} frac{16}{27} Both limits should match for continuity. Correcting the RHS limit, Fleft(frac{2}{3}right) 2 cdot frac{4}{9} - 1 frac{8}{9} - 1 -frac{1}{9} At x 1 Left-hand limit at x 1, F(1) -4 cdot 1 - 1 -5 Right-hand limit at x 1, F(1) -2 cdot 1 - 3 -5 Both limits match, confirming continuity at x 1.

Conclusion

The piecewise function accurately represents the behavior of F(x) |x - 1| - 3|x^2| across the real number line and is continuous everywhere except at the critical points. This method ensures that the function is properly understood and can be plotted accurately by various plotting softwares.

Key Takeaways:

Identify critical points where the modulus expressions change behavior. Break the domain into intervals based on these points. Define the function in each interval by removing modulus. Verify continuity at critical points.