How To Find The Range Of F(x) = |x-5| - 3 A Step-by-Step Guide

Hey everyone! Today, we're diving into the fascinating world of functions, specifically the absolute value function. We're going to figure out the range of the function f(x) = |x - 5| - 3. This might sound intimidating, but trust me, we'll break it down step by step so it's super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Absolute Value Function

Before we tackle the main problem, let's quickly recap what the absolute value function is all about. The absolute value of a number is its distance from zero. Think of it as making everything non-negative. For example, the absolute value of 5 (written as |5|) is 5, and the absolute value of -5 (written as |-5|) is also 5. So, the absolute value function, denoted as |x|, always outputs a non-negative value. This key property is crucial for understanding the range of our function, f(x) = |x - 5| - 3. When dealing with absolute value functions, it's essential to consider how transformations affect the base function, |x|. These transformations, like horizontal shifts, vertical shifts, and reflections, can significantly alter the graph and, consequently, the range. In our case, we have both a horizontal shift (due to the x - 5 inside the absolute value) and a vertical shift (the -3 outside the absolute value). Visualizing these transformations can greatly aid in understanding the final range of the function. We can use graphing tools or simply sketch a graph to see how the basic |x| function is moved and stretched to become f(x) = |x - 5| - 3. This visual representation often makes the range much clearer. Remember, the range is all the possible output values (y-values) of the function, so looking at the graph will tell us the lowest and highest points the function reaches.

Breaking Down f(x) = |x - 5| - 3

Now, let's dissect our function f(x) = |x - 5| - 3. We can think of this function in terms of transformations applied to the basic absolute value function, |x|. First, we have |x - 5|. This represents a horizontal shift. Specifically, it shifts the graph of |x| five units to the right. Think of it this way: the vertex (the pointy part) of the basic absolute value graph |x| is at the origin (0, 0). The vertex of |x - 5| is at x = 5. This is because when x = 5, the expression inside the absolute value becomes |5 - 5| = |0| = 0, which is the minimum value of the absolute value function. The horizontal shift is a crucial concept to grasp because it affects the symmetry of the graph around the new vertex. After the horizontal shift, we have the - 3 part. This represents a vertical shift. It shifts the entire graph down by 3 units. So, the vertex of our transformed function f(x) = |x - 5| - 3 is now at the point (5, -3). This is because the x-value that makes the absolute value part zero is still x = 5, and when x = 5, the function value is f(5) = |5 - 5| - 3 = 0 - 3 = -3. The vertical shift directly impacts the minimum value of the function, which is a key factor in determining the range. Understanding these transformations – the horizontal shift by 5 units and the vertical shift by -3 units – is key to determining the range of the function. By visualizing these shifts, we can see how the basic absolute value function is repositioned in the coordinate plane.

Determining the Range

So, how do we determine the range of f(x) = |x - 5| - 3? Remember, the range is the set of all possible output values (the y-values) that the function can produce. Since the absolute value part, |x - 5|, is always non-negative (it's either zero or positive), the smallest value it can be is 0. This occurs when x = 5. Therefore, the minimum value of |x - 5| is 0. Now, let's consider the entire function, f(x) = |x - 5| - 3. Since the minimum value of |x - 5| is 0, the minimum value of f(x) is 0 - 3 = -3. This means the function will never output a value less than -3. The graph of the function will never go below the horizontal line y = -3. Now, what about the maximum value? As x moves further away from 5 (in either the positive or negative direction), the value of |x - 5| increases. For instance, if x = 10, then |x - 5| = |10 - 5| = 5. If x = 0, then |x - 5| = |0 - 5| = 5. As |x - 5| increases, f(x) = |x - 5| - 3 also increases. There's no upper limit to how large |x - 5| can become, so there's also no upper limit to how large f(x) can become. It can go up to infinity! Therefore, the range of the function f(x) = |x - 5| - 3 includes all real numbers greater than or equal to -3. We can write this mathematically as: R: {f(x) ∈ R | f(x) ≥ -3}. This means the range (R) consists of all f(x) values that belong to the set of real numbers (R) such that f(x) is greater than or equal to -3. So, our function's output will always be -3 or a number bigger than -3.

Visualizing the Range with a Graph

A great way to solidify your understanding is to visualize the function's graph. If you were to plot f(x) = |x - 5| - 3, you'd see a V-shaped graph. The vertex (the bottom point of the V) would be at (5, -3). The V opens upwards, meaning the graph extends upwards indefinitely. This visual representation clearly shows that the lowest point on the graph (the minimum y-value) is -3, and the graph goes up from there. This visually confirms our conclusion about the range: the function takes on all y-values greater than or equal to -3. Graphing the function is an excellent technique for understanding the range and behavior of many different types of functions, not just absolute value functions. It allows you to see the relationship between the input (x) and the output (f(x)) and identify key features like the minimum and maximum values, as well as the overall shape of the function. If you have access to graphing software or a graphing calculator, try plotting f(x) = |x - 5| - 3 yourself. It will undoubtedly enhance your understanding of the range.

Conclusion

So, after our deep dive into f(x) = |x - 5| - 3, we've successfully determined that the range is R: {f(x) ∈ R | f(x) ≥ -3}. We did this by understanding the absolute value function, breaking down the transformations applied to the basic |x| function, and thinking about the possible output values. Remember, the absolute value function always returns a non-negative value, and the transformations (horizontal and vertical shifts) influence the final range. Visualizing the graph is also a super helpful tool. Keep practicing with different functions, and you'll become a range-finding pro in no time! You got this! Understanding the range of a function is a fundamental concept in mathematics, and mastering it opens doors to more advanced topics. By understanding the transformations and the basic properties of the absolute value function, we were able to confidently determine the range. Remember, the key is to break down the function, analyze each component, and then put it all together. And don't forget the power of visualization – graphing can often make complex concepts much easier to grasp.

Therefore, the correct answer is C. R: {f(x) ∈ R | f(x) ≥ -3}

What is the range of the function f(x) = |x-5| - 3? This article provides a comprehensive explanation of how to determine the range of this absolute value function, with detailed steps and explanations.