Finding The Range Of F(x) = -1/3|x-1| - 2

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Hey guys! Today, we're going to dive deep into understanding how to find the range of a function, specifically focusing on the function f(x) = -1/3|x-1| - 2. This might seem a bit tricky at first, but trust me, we'll break it down step by step so it's super clear. Understanding the range of a function is crucial in mathematics as it tells us all the possible output values (y-values) that the function can produce. So, let's get started and unravel this together!

Understanding the Absolute Value Function

Before we jump directly into our function, let's quickly recap the absolute value function. The absolute value of any number is its distance from zero, which is always non-negative. In mathematical terms, |x| is x if x ≥ 0, and -x if x < 0. This |x| part is super important because it forms the backbone of our given function. Think of it like this: no matter what number we plug into the absolute value, we'll always get a positive result (or zero). This is because the absolute value essentially strips away the negative sign if there is one, leaving us with the magnitude of the number. When we look at |x - 1|, we’re shifting this basic absolute value function. The “-1” inside the absolute value means the graph’s vertex (the lowest or highest point) is shifted one unit to the right on the x-axis. This shift is key to understanding the overall behavior of the function and, eventually, its range. Grasping this concept is like having the secret decoder ring for this type of problem!

Analyzing f(x) = -1/3|x-1| - 2

Now, let's break down our function f(x) = -1/3|x-1| - 2 piece by piece. We've already talked about the absolute value part, |x-1|. We know this will always be greater than or equal to zero. But what happens when we multiply it by -1/3? Multiplying by a negative number flips the sign, so our positive absolute value now becomes negative (or zero). More specifically, since |x-1| is always ≥ 0, then -1/3 * |x-1| will always be ≤ 0. This is a critical step! The negative sign in front of the fraction turns the entire term upside down, so to speak. The graph of this function will now open downwards. Next, we subtract 2. This shifts the entire function down by 2 units on the y-axis. Imagine you have the graph of -1/3|x-1|, which peaks at 0. Subtracting 2 simply moves that peak point down to -2. This vertical shift directly impacts our range because it changes the highest possible y-value the function can reach. To summarize, we have an absolute value that's been flipped (due to the negative sign) and shifted downwards. This means our function will have a maximum value, and all other values will be less than or equal to that maximum.

Determining the Maximum Value

To pinpoint the range, we need to find the maximum value of our function. Remember, the absolute value |x-1| is at its minimum when x = 1, which makes |x-1| = 0. This is because the absolute value gives the distance from zero, and the smallest possible distance is zero itself. So, let's plug x = 1 into our function: f(1) = -1/3|1-1| - 2 = -1/3 * 0 - 2 = -2. This tells us that the highest y-value our function can achieve is -2. This is the peak of our function's graph. Any other value of x will result in |x-1| being greater than 0, which, when multiplied by -1/3, will give a negative number. Subtracting 2 from that will make the result even smaller (more negative). Therefore, -2 is indeed the maximum value. We’ve now found the ceiling of our range! The function can go no higher than y = -2. This is a crucial piece of the puzzle, and we’re getting closer to solving it.

Defining the Range

Now that we know the maximum value is -2, we can confidently say that the range of the function f(x) = -1/3|x-1| - 2 includes all real numbers less than or equal to -2. In mathematical notation, we write this as y ≤ -2. This means that the function's output (the y-value) can be -2, or any number smaller than -2. It can go infinitely negative, but it will never be greater than -2. Think of it like a ceiling: the function's values are trapped below -2. To visualize this, imagine the graph of the function. It’s a V-shaped graph (because of the absolute value) that's been flipped upside down and shifted down. The peak of the “V” is at the point (1, -2), and the graph extends downwards from there. This visual representation really helps to solidify the concept of the range. So, there you have it! The range is all the possible y-values, and in this case, they are all less than or equal to -2.

Conclusion

So, the range of the function f(x) = -1/3|x-1| - 2 is all real numbers less than or equal to -2. We figured this out by understanding the absolute value function, analyzing how the different parts of the function transform the graph, and finding the maximum value. Remember, understanding the components of a function – like the absolute value, the negative sign, and the constant term – is key to determining its range. Breaking down the problem into smaller, manageable steps makes it much less daunting. You’ve got this! Keep practicing, and you’ll become a range-finding pro in no time. And remember, math is like a puzzle – challenging, but incredibly rewarding when you solve it. Keep up the great work!