Analyzing The Piecewise Function F(x)

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on a piecewise function named f(x). Piecewise functions are like chameleons, changing their behavior depending on the input value. This particular function, f(x), is defined by three different expressions, each applicable over a specific interval of x. Understanding these functions is crucial in various fields, from calculus to computer science. So, let's put on our mathematical hats and explore this function together!

Dissecting the Definition of f(x)

The function f(x) is defined as follows:

f(x) = { 2^x,  x < 0
       -x^2 - 4x + 1,  0 < x < 2
       1/2 x + 3,  x > 2 }

Breaking it down, we see three distinct 'pieces':

• For x less than 0: The function behaves as an exponential function, specifically 2 raised to the power of x. This part will produce values between 0 and 1 as x approaches negative infinity and 0, respectively.
• For x between 0 and 2: The function transforms into a quadratic expression: -x² - 4x + 1. This is a parabola opening downwards, and its behavior within this interval will be crucial for understanding the overall function.
• For x greater than 2: We encounter a linear function: (1/2)x + 3. This part represents a straight line with a positive slope, meaning the function's value increases as x increases.

Exponential Behavior: x < 0

Let's zoom in on the first piece: f(x) = 2^x for x < 0. This is an exponential decay function. Exponential functions are characterized by rapid growth or decay, and in this case, we have decay because the base (2) is greater than 1, and the exponent is negative. As x becomes more and more negative, 2^x gets closer and closer to 0, but it never actually reaches 0. This behavior creates a horizontal asymptote at y = 0. Understanding exponential functions is key to grasping phenomena like radioactive decay and compound interest. We can analyze this behavior further by considering some specific values. For instance, when x = -1, f(x) = 2^(-1) = 0.5. When x = -2, f(x) = 2^(-2) = 0.25. Notice how the value halves each time x decreases by 1. This is the essence of exponential decay.

Quadratic Territory: 0 < x < 2

Now, let's explore the second piece: f(x) = -x² - 4x + 1 for 0 < x < 2. This is a quadratic function, represented by a parabola. The negative coefficient of the x² term (-1) tells us that the parabola opens downwards, meaning it has a maximum point. To fully understand this quadratic, we need to find its vertex (the maximum point) and its behavior within the interval 0 < x < 2. We can find the x-coordinate of the vertex using the formula x = -b / 2a, where a and b are the coefficients of the quadratic expression. In this case, a = -1 and b = -4, so the x-coordinate of the vertex is x = -(-4) / (2 * -1) = -2. However, this vertex is outside our interval of 0 < x < 2. Therefore, we need to analyze the function's values at the boundaries of the interval and within it. At x = 0 (approaching from the right), f(x) approaches 1. At x = 2 (approaching from the left), f(x) = -(2)² - 4(2) + 1 = -4 - 8 + 1 = -11. This indicates that the function decreases sharply within this interval. Quadratic functions are ubiquitous in physics and engineering, describing trajectories of projectiles and the shapes of suspension cables.

Linear Ascent: x > 2

Finally, let's examine the third piece: f(x) = (1/2)x + 3 for x > 2. This is a linear function, represented by a straight line. The slope of this line is 1/2, which is positive, indicating that the function increases as x increases. The y-intercept is 3, but this point is not relevant since we are only considering x > 2. At x = 2 (approaching from the right), f(x) approaches (1/2)(2) + 3 = 4. As x continues to increase, the function will increase linearly. Linear functions are the simplest type of functions and are used extensively to model direct relationships between variables. In economics, for example, they might represent cost functions or demand curves.

Key Statements and Analysis

To determine which statements are true about this function, we need to consider its behavior across the entire domain. Here's a breakdown of some potential statements and how we might analyze them: