Analyzing The Piecewise Function F(x) A Comprehensive Guide

Hey guys! Let's dive into the fascinating world of piecewise functions. Today, we're going to break down a specific function, $f(x)$, and explore its behavior across different intervals. Piecewise functions might seem a bit intimidating at first, but trust me, once you get the hang of them, they're super cool and useful in various areas of mathematics and beyond!

Defining the Function f(x)

So, what exactly is this function $f(x)$ we're talking about? Well, it's defined in pieces, meaning it has different formulas for different ranges of $x$ values. Here's the breakdown:

• For x < 0, $f(x) = 2^x$
• For 0 < x < 2, $f(x) = -x^2 - 4x + 1$
• For x > 2, $f(x) = (1/2)x + 3

Notice how the function's definition changes depending on the value of $x$. This is the essence of a piecewise function. Each piece has its own domain, and the function behaves differently within each domain. Understanding these pieces and how they connect is key to understanding the overall behavior of the function.

Analyzing the Pieces

Let's take a closer look at each piece individually. This will help us build a complete picture of $f(x)$.

Piece 1: $f(x) = 2^x$ for $x < 0$

This part of the function is an exponential function. Remember those? The base is 2, and the exponent is $x$. When $x$ is negative, we're dealing with fractional powers of 2. As $x$ gets more and more negative, $2^x$ gets closer and closer to 0, but it never actually reaches 0. This means we have a horizontal asymptote at $y = 0$ for this piece. Also, when x approaches zero from the left side, f(x) approaches 1. The graph of this piece will start close to the x-axis on the left and rise towards the point (0,1). It's a classic exponential decay curve as we move from left to right.

Piece 2: $f(x) = -x^2 - 4x + 1$ for $0 < x < 2$

This piece is a quadratic function, and it's got a negative leading coefficient (-1), which means it's a parabola that opens downwards. Parabolas are always fun! To get a good handle on this piece, we can find its vertex. The x-coordinate of the vertex is given by $-b/2a$, where $a$ and $b$ are the coefficients in the quadratic equation. In this case, $a = -1$ and $b = -4$, so the x-coordinate of the vertex is $-(-4) / (2 * -1) = -2$. However, this vertex is outside of our domain $0 < x < 2$, meaning that the maximum of this function in the specified domain is not at its vertex. To determine the behavior within our domain, we can evaluate the function at the endpoints of the interval, excluding the endpoints themselves as our domain is strictly between 0 and 2. Evaluating at x approaching 0 gives f(x) approaching 1, and evaluating at x approaching 2 gives f(x) approaching -11. This parabola opens downwards, and our piece covers a portion of the curve that is decreasing. The vertex of the complete parabola would have been at x = -2, which is not in our interval, but tells us that over our domain the function is decreasing, as we are to the right of the vertex.

Piece 3: $f(x) = (1/2)x + 3$ for $x > 2$

This piece is a linear function with a slope of 1/2 and a y-intercept of 3. Straight lines are the simplest functions, right? This piece is a line that slopes upwards as $x$ increases. To visualize this, we can think about two points on the line. As x approaches 2 from the right side, f(x) approaches 4. As x increases beyond 2, the line continues upwards with a gentle slope of 1/2. For each 2 units we move to the right, the line goes up by 1 unit.

Putting it All Together

Now that we've analyzed each piece individually, let's think about how they all fit together to form the complete function $f(x)$. We have three distinct sections:

1. An exponential decay curve for $x < 0$
2. A downward-sloping portion of a parabola for $0 < x < 2$
3. An upward-sloping line for $x > 2

The key thing to remember with piecewise functions is that they can have discontinuities. This means there might be jumps or breaks in the graph where the pieces connect. For instance, let's examine the point where the first and second pieces meet, as x approaches 0. From the left side (x < 0), f(x) approaches 1. From the right side (0 < x < 2), f(x) also approaches 1. At the point where the second and third pieces meet, as x approaches 2, from the left side (0 < x < 2), f(x) approaches -11. As x approaches 2 from the right side (x > 2), f(x) approaches 4. Because these limits do not match, the function is discontinuous at x = 2.

Analyzing the Truth of Statements About Function f(x)

Alright, now that we have a solid understanding of the function $f(x)$, let's consider what kind of statements we can make about it and determine their truthfulness. Typical statements might involve:

• Continuity: Is the function continuous everywhere? As we discussed earlier, discontinuities can occur at the boundaries between pieces.
• Range: What are the possible output values of the function? We need to consider the range of each piece and how they combine.
• Monotonicity: Is the function increasing or decreasing over certain intervals? We can analyze the slope of each piece to determine this.
• Boundedness: Is the function bounded above or below? Does it approach infinity or negative infinity as x goes to extremes?

Evaluating Continuity

To check for continuity, we need to ensure that the function pieces connect smoothly at the boundaries of their domains. This means the left-hand limit and the right-hand limit must be equal at these points, and that those limits also equal the function's value at that point, if the point is included in the domain. We've already seen that this doesn't hold at x = 2, so the function is discontinuous there.

Determining the Range

To figure out the range, we need to look at the output values each piece can produce. The exponential piece, $2^x$ for $x < 0$, outputs values between 0 (exclusive) and 1. The quadratic piece, $-x^2 - 4x + 1$ for $0 < x < 2$, outputs values between -11 and 1. The linear piece, $(1/2)x + 3$ for $x > 2$, outputs values greater than 4. Combining these, the range is all real numbers greater than -11. Note that -11 is not included because x cannot equal 2 in the second piece, however, there are no maximum bounds to the range as the linear function continues to increase infinitely.

Assessing Monotonicity

Monotonicity refers to whether the function is increasing or decreasing. The exponential piece is increasing. The quadratic piece, in our domain, is decreasing. The linear piece is increasing. So, the function has both increasing and decreasing intervals.

Checking Boundedness

Boundedness refers to whether the function's output values are limited. Our function is bounded below because there's a lower limit to the quadratic piece. However, it's not bounded above because the linear piece goes to infinity as x increases. Because of the linear piece increasing infinitely, the exponential piece decaying to zero, and the quadratic portion having a lower bound, we know the function is bounded below.

Conclusion

Piecewise functions are a powerful tool for modeling situations where different rules apply under different conditions. By carefully analyzing each piece and how they connect, we can gain a deep understanding of the function's behavior. Remember to pay attention to continuity, range, monotonicity, and boundedness, and you'll be well-equipped to tackle any piecewise function that comes your way! Understanding the function's pieces and their relationship to each other gives us an insight into the entire range, domain, and continuity of the overall function. You've got this!